******************************************************* **** Welcome to the PDE2D 9.6 Interactive Driver **** ******************************************************* PDE2D can solve systems of (linear or nonlinear) steady-state, time-dependent and eigenvalue partial differential equations in 1D intervals, general 2D regions, and in a wide range of simple 3D regions. Ordinary differential equation systems can also be solved. You will now be asked a series of interactive questions about your problem. The answers you give will be used to construct a PDE2D FORTRAN program, which can then be compiled and linked with the PDE2D runtime routines to produce an executable program. The FORTRAN driver program created will be well-documented and highly readable (most of the interactive messages are repeated in the comments), so that minor modifications or corrections can be made directly to the FORTRAN program, without the need to work through a new interactive session. You can alternatively create your PDE2D FORTRAN program using the PDE2D graphical user interface (GUI) ("pde2d_gui [progname]"). It [RETURN] is extraordinarily easy to set up problems using the PDE2D GUI, which handles 0D and 1D problems, and 2D and 3D problems in "a wide range of simple regions". However, the PDE2D GUI cannot handle complex regions, so if you have a complex 2D region you must use this Interactive Driver. If this is your first time to use PDE2D, you may want to work through an example problem before trying one of your own. Do you want to work through a prepared example? |---- Enter yes or no yes Several prepared examples are available. Enter: 1 - to see a simple problem: a simply-supported elastic plate equation, with a unit load concentrated at the midpoint of a square. 2 - to see a more complex problem: a non-linear, steady-state PDE, solved in an annulus. Dirichlet (U = ...) boundary conditions are imposed on part of the boundary, and Neumann (dU/dn = ...) conditions are imposed on the other part, in this example. The initial triangulation is generated automatically, and adaptive grid refinement is illustrated. 3 - to see an eigenvalue problem. The region has a curved interface across which material properties vary abruptly, in this example. 4 - to see the first part of a thermal stress problem. In this part, the temperature distribution in a V-notched block is calculated by solving the time-dependent heat conduction equation, using adaptive time step control. 5 - to see the second part of a thermal stress problem. In this part, the stresses induced in the V-notched block by thermal expansion are calculated, using the temperature distribution output by example 4. You must run example 4 [RETURN] and save the tabular output before you can run example 5. Examples 4 and 5 illustrate communication between problems. 6 - to see a 1D time-dependent integro-differential equation for a financial math application. In this problem there is a term involving an integral of the solution, which requires that we use PDE2D's feature for interpolating the solution at the last saved time step, for use in the integral term. 7 - to see a waveguide problem (an eigenvalue problem in which the eigenvalue appears nonlinearly). This example shows how to handle boundary conditions of different types on the same arc, and how to produce a plot of a computed integral vs time. 8 - to see the Navier-Stokes equations (penalty formulation) solved for a fluid flowing around a bend. 9 - to see a 3D elasticity problem, solved in a torus. This example illustrates the use of user-defined coordinate transformations to handle more general 3D regions. 10 - to see a time-dependent wave equation (reduced to a system of two PDEs), solved in a 3D box. 11 - to see a 3D eigenvalue problem (the Schrodinger equation in a hydrogen atom). This example illustrates the use of spherical coordinates and periodic boundary conditions. 12 - to see a 3D eigenvalue problem, solved in a composite region [RETURN] consisting of two cylinders of different material properties. 13 - to see the axisymmetric Navier-Stokes equations solved in a non-rectangular channel, using the collocation FEM. 14 - to see a 1D saturated/unsaturated water flow problem. 15 - to see a 1D version of the Schrodinger eigenvalue equation of example 11. 0 - (no example) If you select one of the examples, the correct answer for each interactive question will be supplied after the question. |---- Enter an integer value in the range 0 to 15 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ In this example, the propagating modes (eigenvalues) of a glass $ $ waveguide embedded in a silica substrate are to be calculated. $ $ The equations are $ $ $ $ (E2)x + (-E1)y + w*amu*H = 0 $ $ (-H2)x + (H1)y + w*eps*E = 0 $ $ where $ $ E1 = (-w*amu*Hy - beta*Ex)/(w**2*amu*eps - beta**2) $ $ E2 = (w*amu*Hx - beta*Ey)/(w**2*amu*eps - beta**2) $ $ H1 = (w*eps*Ey - beta*Hx)/(w**2*amu*eps - beta**2) $ $ H2 = (-w*eps*Ex - beta*Hy)/(w**2*amu*eps - beta**2) $ $ and $ $ H,E = z-components of magnetic and electric fields $ $ w = 2*pi*c/lambda (pi=3.14159; c=2.9979 E8) $ $ lambda = 0.85 E-6 (wavelength) $ $ amu = 12.566 E-7 (permeability) $ $ eps = 8.854 E-12*n(x,y)**2 (permittivity) $ $ n(x,y) = 1.55 in glass (see figure below) $ $ = 1.50 in silica (see figure below) $ $ beta = eigenvalue (propagating mode) $ $ $ [RETURN] $ 4 -------------------------------- $ $ | | $ $ | | $ $ 1 | -------- | $ $ | | | | $ $ | | glass| silica | $ $ | | | | $ $ -1 | -------- | $ $ | | $ $ | | $ $ -4 -------------------------------- $ $ -4 -1 1 4 (micrometers) $ $ $ $ The boundary conditions are dH/dn=0, E=0 on the entire boundary. $ $ $ $ This is a linear PDE, but since the eigenvalue beta appears in a $ $ nonlinear form, it does not fit into the standard PDE2D eigenvalue $ $ problem format, so a different approach will be taken. Some $ $ "random" nonhomogeneous terms will be added to each partial $ $ differential equation, and the problem will be solved for various $ $ values of beta in a range (beta0,beta1), that is, with beta = $ $ beta0 + i/NSTEPS*(beta1-beta0), i=1,...,NSTEPS. When beta is close $ [RETURN] $ to an eigenvalue, the corresponding solution will be very large. $ $ It is known that there is an eigenvalue at beta = 1.134 E7. $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ [RETURN] In what follows, when you are told to enter a 'FORTRAN expression', this means any valid FORTRAN expression of 65 characters or less. In this expression, you may include references to FORTRAN function subprograms. You may define these functions line by line at the end of the interactive session, when prompted, or add them later using an editor. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you enter a "#" in the first column of any input line, this + + instructs the interactive driver to read this and subsequent input + + lines from the file "pde2d.in". A "#" in the first column of an + + input line in the file "pde2d.in" (or an end-of-file) instructs the + + driver to switch back to interactive input. + + + + All lines input during an interactive session are echo printed to + + a file "echo.out". You may want to modify this file and rename it + + "pde2d.in", and read some or all of your input from this file during + + your next interactive session. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ PDE2D can solve problems with 0,1,2 or 3 space variables. Enter the dimension of your problem: [RETURN] 0 - to solve a time-dependent ordinary differential equation system, or an algebraic or algebraic eigenvalue system 1 - to solve problems in 1D intervals 2 - to solve problems in general 2D regions 3 - to solve problems in a wide range of simple 3D regions $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: 2 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter an integer value in the range 0 to 3 2 Which finite element method do you want to use: 1. Galerkin method 2. Collocation method The collocation method can handle a wide range of simple 2D regions; the Galerkin method can handle completely general 2D regions. Enter 1 or 2 to select a finite element method. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you have a composite region, with discontinuous material + + parameters, you should use the Galerkin method. If your partial + + differential equations and boundary conditions are difficult to put + + into the "divergence" form required by the Galerkin method, or if + + you have periodic boundary conditions, use the collocation method. + + The collocation method produces an approximate solution with + + continuous first derivatives; the Galerkin solution is continuous + + but its first derivatives are not. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: 1 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ [RETURN] |---- Enter an integer value in the range 1 to 2 1 Is double precision mode to be used? Double precision is recommended. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If double precision mode is used, variables and functions assigned + + names beginning with a letter in the range A-H or O-Z will be DOUBLE + + PRECISION, and you should use double precision constants and FORTRAN + + expressions throughout; otherwise such variables and functions will + + be of type REAL. In either case, variables and functions assigned + + names beginning with I,J,K,L,M or N will be of INTEGER type. + + + + It is possible to convert a single precision PDE2D program to double + + precision after it has been created, using an editor. Just change + + all occurrences of "real" to "double precision" + + " tdp" to "dtdp" (note leading blank) + + Any user-written code or routines must be converted "by hand", of + + course. To convert from double to single, reverse the changes. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: yes $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no yes If you don't want to read the FINE PRINT, default NPROB. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you want to solve several similar problems in the same run, set + + NPROB equal to the number of problems you want to solve. Then NPROB + + loops through the main program will be done, with IPROB=1,...,NPROB, + + and you can make the problem parameters vary with IPROB. NPROB + + defaults to 1. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ press [RETURN] to default NPROB $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NPROB = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| PDE2D solves the time-dependent system (note: U,A,B,F,FB,GB,U0 may be vectors, C,RHO may be matrices): C(X,Y,T,U,Ux,Uy)*d(U)/dT = d/dX* A(X,Y,T,U,Ux,Uy) + d/dY* B(X,Y,T,U,Ux,Uy) - F(X,Y,T,U,Ux,Uy) or the steady-state system: d/dX* A(X,Y,U,Ux,Uy) + d/dY* B(X,Y,U,Ux,Uy) = F(X,Y,U,Ux,Uy) or the linear and homogeneous eigenvalue system: d/dX* A(X,Y,U,Ux,Uy) + d/dY* B(X,Y,U,Ux,Uy) = F(X,Y,U,Ux,Uy) + lambda*RHO(X,Y)*U in an arbitrary two-dimensional region, R, with 'fixed' boundary conditions on part of the boundary: [RETURN] U = FB(X,Y,[T]) and 'free' boundary conditions on the other part: A*nx + B*ny = GB(X,Y,[T],U,Ux,Uy) For time-dependent problems there are also initial conditions: U = U0(X,Y) at T=T0 Here Ux,Uy represent the (vector) functions dU/dX,dU/dY, and (nx,ny) represents the unit outward normal to the boundary. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If your PDEs involve the solution at points other than (X,Y), the + + function + + (D)OLDSOL2(IDER,IEQ,XX,YY,KDEG) + + will interpolate (using interpolation of degree KDEG=1,2 or 3) to + + (XX,YY) the function saved in UOUT(*,*,IDER,IEQ,ISET) on the last + + time step or iteration (ISET) for which it has been saved. Thus, + + for example, if IDER=1, this will return the latest value of + + component IEQ of the solution at (XX,YY), assuming this has not been + [RETURN] + modified using UPRINT... If your equations involve integrals of the + + solution, for example, you can use (D)OLDSOL2 to approximate these + + using the solution from the last time step or iteration. + + + + CAUTION: For a steady-state or eigenvalue problem, you must reset + + NOUT=1 if you want to save the solution each iteration. + ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + A system of NEQN complex partial differential equations must be + + written as a system of 2*NEQN real equations, by separating the + + equations into their real and imaginary parts. However, note that + + the complex arithmetic abilities of FORTRAN can be used to simplify + + this separation. For example, the complex PDE: + + I*(Uxx+Uyy) = 1/(1+U**10), where U = UR + UI*I + + would be difficult to split up analytically, but using FORTRAN + + expressions it is easy: + + A1 = -UIx, B1 = -UIy, F1 = REAL(1.0/(1.0+CMPLX(UR,UI)**10)) + + A2 = URx, B2 = URy, F2 = AIMAG(1.0/(1.0+CMPLX(UR,UI)**10)) + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ You may now define global parameters, which may be referenced in any of the "FORTRAN expressions" you input throughout the rest of this interactive session. You will be prompted alternately for parameter names and their values; enter a blank name when you are finished. [RETURN] Parameter names are valid FORTRAN variable names, starting in column 1. Thus each name consists of 1 to 6 alphanumeric characters, the first of which must be a letter. If the first letter is in the range I-N, the parameter must be an integer. Parameter values are either FORTRAN constants or FORTRAN expressions involving only constants and global parameters defined on earlier lines. They may also be functions of the problem number IPROB, if you are solving several similar problems in one run (NPROB > 1). Note that you are defining global CONSTANTS, not functions; e.g., parameter values may not reference any of the independent or dependent variables of your problem. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you define other parameters here later, using an editor, you must + + add them to COMMON block /PARM8Z/ everywhere this block appears, if + + they are to be "global" parameters. + + + + The variable PI is already included as a global parameter, with an + + accurate value 3.14159... + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ [RETURN] $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: C $ $ 2.9979 E8 $ $ W $ $ 2*PI*C/0.85E-6 $ $ AMU $ $ 12.566 E-7 $ $ [blank line] $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ Parameter name = (type blank line to terminate) C C = |----Enter constant or FORTRAN expression-----------------------| 2.9979 E8 Parameter name = (type blank line to terminate) W W = |----Enter constant or FORTRAN expression-----------------------| 2*PI*C/0.85 E-6 Parameter name = (type blank line to terminate) AMU AMU = |----Enter constant or FORTRAN expression-----------------------| 12.566 E-7 Parameter name = (type blank line to terminate) If you don't want to read the FINE PRINT, enter 'no'. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Do you want to be given a chance to write a FORTRAN block before the + + definitions of many functions? If you answer 'no', you will still + + be given a chance to write code before the definition of the PDE + + coefficients, but not other functions. Of course, you can always + + add code later directly to the resulting program, using an editor. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no Now, the first step in creating a 2D PDE2D program is to construct an initial triangulation of the region R over which the partial differential equations are to be solved. This initial triangulation can later be refined and graded to your specifications. The boundary of the region R should be divided into distinct (curved or straight) arcs, each of which is smooth with smooth boundary conditions. Thus at every corner or point where the boundary conditions have a discontinuity or change type, a new boundary arc should begin. Each arc is assigned a unique integer arc number, which is arbitrary except that it must be negative if 'fixed' boundary conditions are specified on that arc, and positive (but < 1000) if 'free' boundary conditions are specified. 'Fixed' means that all unknowns are specified on that boundary (Dirichlet type conditions) and 'free' refers to more general boundary conditions, as defined earlier. These more general conditions include specification of the boundary flux (or normal derivative in the case of a Laplacian operator). +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If some, but not all, of the unknowns are specified on a boundary + [RETURN] + arc, that arc should be considered 'free', since even 'fixed' + + type conditions of the form: + + Ui = FBi(X,Y,[T]) + + can be expressed as 'free' boundary conditions in the form: + + Ai*nx + Bi*ny = zero(Ui-FBi(X,Y,[T])) + + where zero(f) = Big_Number*f is a PDE2D-supplied function. + ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + If Ai=Bi=0 (eg, if the PDE is first order), then setting GBi=0 is + + equivalent to setting "no" boundary condition (which is sometimes + + appropriate for first order PDEs), because the boundary condition + + Ai*nx + Bi*ny = GBi reduces to 0=0. + ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + Note: while 'fixed' boundary conditions are enforced exactly (at the + + nodes), 'free' boundary conditions are not; the greater the overall + + solution accuracy, the more closely they are satisfied. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ There are 3 options for generating the initial triangulation: 1. If the region R is a rectangle with sides parallel to the X and Y axes, then the initial triangulation can be generated automatically. 2. If the region R can be conveniently described by parametric [RETURN] equations in the form: X = X(P,Q) Y = Y(P,Q) where P and Q have constant limits (for example, if X=P*COS(Q), Y=P*SIN(Q), P and Q will represent polar coordinates), then the initial triangulation can also be generated automatically. 3. For more general regions, you will need to create an initial triangulation by hand. Enter INTRI = 1,2 or 3 to choose an initial triangulation option. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ Since the boundary conditions are mixed, and the same on all arcs, $ $ we assign a positive number (1) to each arc, and since the region $ $ is a square, $ $ enter: INTRI = 1 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ INTRI = |---- Enter an integer value in the range 1 to 3 1 For a rectangular region, the initial triangulation is defined by a set of grid lines corresponding to X = XGRID(1),XGRID(2),...,XGRID(NXGRID) Y = YGRID(1),YGRID(2),...,YGRID(NYGRID) Each grid rectangle is divided into four equal area triangles. You will first be prompted for NXGRID, the number of X-grid points, then for XGRID(1),...,XGRID(NXGRID). Any points defaulted will be uniformly spaced between the points you define; the first and last points represent the values of X on the left and right sides of the rectangle R, and cannot be defaulted. Then you will be prompted similarly for the number and values of the Y-grid points. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: NXGRID = 4 $ $ It is important for accuracy considerations that no triangles in $ $ the initial triangulation (hence none in the final triangulation) $ $ straddle the interface between the glass and silica, so enter: $ $ XGRID(1) = -4.E-6 $ $ XGRID(2) = -1.E-6 $ $ XGRID(3) = 1.E-6 $ $ XGRID(NXGRID) = 4.E-6 $ $ similarly, $ [RETURN] $ enter: NYGRID = 4 $ $ YGRID(1) = -4.E-6 $ $ YGRID(2) = -1.E-6 $ $ YGRID(3) = 1.E-6 $ $ YGRID(NYGRID) = 4.E-6 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NXGRID = |---- Enter an integer value in the range 2 to +INFINITY 4 XGRID(1) = |----Enter constant or FORTRAN expression-----------------------| -4.E-6 XGRID(2) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| -1.E-6 XGRID(3) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| 1.E-6 XGRID(NXGRID) = |----Enter constant or FORTRAN expression-----------------------| 4.E-6 NYGRID = |---- Enter an integer value in the range 2 to +INFINITY 4 YGRID(1) = |----Enter constant or FORTRAN expression-----------------------| -4.E-6 YGRID(2) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| -1.E-6 YGRID(3) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| 1.E-6 YGRID(NYGRID) = |----Enter constant or FORTRAN expression-----------------------| 4.E-6 Enter the arc numbers IXARC(1),IXARC(2) of the left and right sides, respectively, and the arc numbers IYARC(1),IYARC(2) of the bottom and top sides of the rectangle R. Recall that negative arc numbers correspond to 'fixed' boundary conditions, and positive arc numbers ( < 1000) correspond to 'free' boundary conditions. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: IXARC(1) = 1 IXARC(2) = 1 $ $ IYARC(1) = 1 IYARC(2) = 1 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ IXARC(1) = |----Enter constant or FORTRAN expression-----------------------| 1 IXARC(2) = |----Enter constant or FORTRAN expression-----------------------| 1 IYARC(1) = |----Enter constant or FORTRAN expression-----------------------| 1 IYARC(2) = |----Enter constant or FORTRAN expression-----------------------| 1 How many triangles (NTF) are desired for the final triangulation? $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: NTF = 100 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NTF = |---- Enter an integer value in the range 36 to +INFINITY 100 Enter a FORTRAN expression for TRIDEN(X,Y), which controls the grading of the triangulation. TRIDEN should be largest where the triangulation is to be most dense. The default is TRIDEN(X,Y)=1.0 (a uniform triangulation). TRIDEN may also be a function of the initial triangle number KTRI. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: TRIDEN = EXP(-(X/1.E-6)**2-(Y/1.E-6)**2) $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ TRIDEN = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| EXP(-(X/1.E-6)**2-(Y/1.E-6)**2) If you don't want to read the FINE PRINT, enter 'no'. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Do you want the triangulation to be graded adaptively? + + + + If you answer "yes", make sure there is no "pde2d.adp" file in the + + working directory the first time you run the program, then run the + + program two or more times, possibly increasing NTF each time. On + + the first run, the triangulation will be graded as guided by + + TRIDEN(X,Y), but on each subsequent run, information output by the + + previous run to "pde2d.adp" will be used to guide the grading of the + + new triangulation. + + + If ADAPT=.TRUE., after each run a file "pde2d.adp" is written + + which tabulates the values of the magnitude of the gradient of the + + solution (at the last time step or iteration) at an output NXP8Z by + + NYP8Z grid of points (NXP8Z and NYP8Z are set to 101 in a PARAMETER + + statement in the main program, so they can be changed if desired). + + If NEQN > 1, a normalized average of the gradients of the NEQN + + solution components is used. + + + + You can do all the "runs" in one program, by setting NPROB > 1. + [RETURN] + Each pass through the DO loop, PDE2D will read the gradient values + + output the previous pass. If RESTRT=.TRUE., GRIDID=.FALSE., and + + T0,TF are incremented each pass through the DO loop, it is possible + + in this way to solve a time-dependent problem with an adaptive, + + moving, grid. + + + + Increase the variable EXAG from its default value of 1.5 if you want + + to exaggerate the grading of an adaptive triangulation (make it less + + uniform). EXAG should normally not be larger than about 2.0. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ If you don't want to read the FINE PRINT, enter 'no'. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no If you don't want to read the FINE PRINT, default SHAPE. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Enter a FORTRAN expression for SHAPE(X,Y), which controls the + + approximate shape of the triangles. The triangulation refinement + + will proceed with the goal of generating triangles with an average + + height to width ratio of approximately SHAPE(X,Y) near the point + + (X,Y). SHAPE must be positive. The default is SHAPE(X,Y)=1.0. + + + + SHAPE may also be a function of the initial triangle number KTRI. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ press [RETURN] to default SHAPE $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ SHAPE = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| If you don't want to read the FINE PRINT, enter ISOLVE = 4. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + The following linear system solvers are available: + + + + 1. Band method + + The band solver uses a reverse Cuthill-McKee ordering. + + 2. Frontal method + + This is an out-of-core version of the band solver. + + 3. Jacobi bi-conjugate gradient method + + This is a preconditioned bi-conjugate gradient, or + + Lanczos, iterative method. (This solver is MPI- + + enhanced, if MPI is available.) If you want to + + override the default convergence tolerance, set a + + new relative tolerance CGTL8Z in the main program. + + 4. Sparse direct method + + This is based on Harwell Library routines MA27/MA37, + + developed by AEA Industrial Technology at Harwell + + Laboratory, Oxfordshire, OX11 0RA, United Kingdom + + (used by permission). + + 5. Local solver + + Choose this option ONLY if alternative linear system + [RETURN] + solvers have been installed locally. See subroutines + + (D)TD3M, (D)TD3N in file (d)subs.f for instructions + + on how to add local solvers. + + 6. MPI-based parallel band solver + + This is a parallel solver which runs efficiently on + + multiple processor machines, under MPI. It is a + + band solver, with the matrix distributed over the + + available processors. Choose this option ONLY if the + + solver has been activated locally. See subroutine + + (D)TD3O in file (d)subs.f for instructions on how to + + activate this solver and the MPI-enhancements to the + + conjugate gradient solver. + + + + Enter ISOLVE = 1,2,3,4,5 or 6 to select a linear system solver. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ If you don't want to read the FINE PRINT, enter ISOLVE = 4. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: ISOLVE = 4 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ISOLVE = |---- Enter an integer value in the range 1 to 6 4 Enter the element degree (1,2,3 or 4) desired. A suggested value is IDEG = 3. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + A negative value for IDEG can be entered, and elements of degree + + ABS(IDEG) will be used, with a lower order numerical integration + + scheme. This results in a slight increase in speed, but negative + + values of IDEG are normally not recommended. + + + + The spatial discretization error is O(h**2), O(h**3), O(h**4) or + + O(h**5) when IDEG = 1,2,3 or 4, respectively, is used, where h is + + the maximum triangle diameter, even if the region is curved. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: IDEG = 3 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ IDEG = |---- Enter an integer value in the range -4 to 4 3 What type of PDE problem do you want to solve? 1. a steady-state (time-independent) problem 2. a time-dependent problem 3. a linear, homogeneous eigenvalue problem Enter 1,2 or 3 to select a problem type. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ Although we want to solve a series of steady-state problems, it is $ $ more convenient to consider this as a time-dependent problem, with $ $ C11=C12=C21=C22=0 and with the time variable T used to represent the $ $ parameter beta. Thus $ $ enter: 2 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter an integer value in the range 1 to 3 2 Enter the initial time value (T0) and the final time value (TF), for this time-dependent problem. T0 defaults to 0. TF is not required to be greater than T0. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ We want to search the interval 1.13 E7 < beta (=T) < 1.14 E7 for $ $ eigenvalues, so $ $ enter: T0 = 1.13E7 $ $ TF = 1.14E7 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ T0 = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| 1.13E7 TF = |----Enter constant or FORTRAN expression-----------------------| 1.14E7 Is this a linear problem? ("linear" means all differential equations and all boundary conditions are linear). If you aren't sure, it is safer to answer "no". $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: yes $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no yes Do you want the time step to be chosen adaptively? If you answer 'yes', you will then be prompted to enter a value for TOLER(1), the local relative time discretization error tolerance. The default is TOLER(1)=0.01. If you answer 'no', a user-specified constant time step will be used. We suggest that you answer 'yes' and default TOLER(1) (although for certain linear problems, a constant time step may be much more efficient). +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If a negative value is specified for TOLER(1), then ABS(TOLER(1)) is + + taken to be the "absolute" error tolerance. If a system of PDEs is + + solved, by default the error tolerance specified in TOLER(1) applies + + to all variables, but the error tolerance for the J-th variable can + + be set individually by specifying a value for TOLER(J) using an + + editor, after the end of the interactive session. + + + + Each time step, two steps of size dt/2 are taken, and that solution + + is compared with the result when one step of size dt is taken. If + + the maximum difference between the two answers is less than the + + tolerance (for each variable), the time step dt is accepted (and the + + next step dt is doubled, if the agreement is "too" good); otherwise + + dt is halved and the process is repeated. Note that forcing the + [RETURN] + local (one-step) error to be less than the tolerance does not + + guarantee that the global (cumulative) error is less than that value.+ + However, as the tolerance is decreased, the global error should + + decrease correspondingly. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no If you don't want to read the FINE PRINT, it is safe (though possibly very inefficient) to enter 'no'. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If your time-dependent problem is linear with all PDE and boundary + + condition coefficients independent of time except inhomogeneous + + terms, then a large savings in execution time may be possible if + + this is recognized (the LU decomposition computed on the first step + + can be used on subsequent steps). Is this the case for your + + problem? (Caution: if you answer 'yes' when you should not, you + + will get incorrect results with no warning.) + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no The time stepsize will be constant, DT = (TF-T0)/NSTEPS. Enter a value for NSTEPS, the number of time steps. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you later turn on adaptive step control, the time stepsize will be+ + chosen adaptively, between an upper limit of DTMAX = (TF-T0)/NSTEPS + + and a lower limit of 0.0001*DTMAX. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ We will try 100 values of beta (=T) in the interval (T0,TF), so $ $ enter: NSTEPS = 100 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NSTEPS = |----Enter constant or FORTRAN expression-----------------------| 100 If you don't want to read the FINE PRINT, enter 'no'. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Is the Crank-Nicolson scheme to be used to discretize time? If you + + answer 'no', a backward Euler scheme will be used. + + + + If a user-specified constant time step is chosen, the second order + + Crank Nicolson method is recommended only for problems with very + + well-behaved solutions, and the first order backward Euler scheme + + should be used for more difficult problems. In particular, do not + + use the Crank Nicolson method if the left hand side of any PDE is + + zero, for example, if a mixed elliptic/parabolic problem is solved. + + + + If adaptive time step control is chosen, however, an extrapolation + + is done between the 1-step and 2-step answers which makes the Euler + + method second order, and the Crank-Nicolson method strongly stable. + + Thus in this case, both methods have second order accuracy, and both + + are strongly stable. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ Since C11=C12=C21=C22=0, it is essential that the backward Euler $ $ method be used, so $ [RETURN] $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no How many differential equations (NEQN) are there in your problem? $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: NEQN = 2 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NEQN = |---- Enter an integer value in the range 1 to 99 2 You may now choose names for the component(s) of the (possibly vector) solution U. Each must be an alphanumeric string of one to three characters, beginning with a letter in the range A-H or O-Z. The variable names X,Y,T,S,A and B must not be used. The name should start in column 1. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: U1 = H $ $ U2 = E $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ U1 = H U2 = E PDE2D solves the system of equations: C11(X,Y,T,H,Hx,Hy,E,Ex,Ey)*d(H)/dT + C12(X,Y,T,H,Hx,Hy,E,Ex,Ey)*d(E)/dT = d/dX* A1(X,Y,T,H,Hx,Hy,E,Ex,Ey) + d/dY* B1(X,Y,T,H,Hx,Hy,E,Ex,Ey) - F1(X,Y,T,H,Hx,Hy,E,Ex,Ey) C21(X,Y,T,H,Hx,Hy,E,Ex,Ey)*d(H)/dT + C22(X,Y,T,H,Hx,Hy,E,Ex,Ey)*d(E)/dT = d/dX* A2(X,Y,T,H,Hx,Hy,E,Ex,Ey) + d/dY* B2(X,Y,T,H,Hx,Hy,E,Ex,Ey) - F2(X,Y,T,H,Hx,Hy,E,Ex,Ey) with 'fixed' boundary conditions: H = FB1(X,Y,T) E = FB2(X,Y,T) or 'free' boundary conditions: A1*nx + B1*ny = GB1(X,Y,T,H,Hx,Hy,E,Ex,Ey) A2*nx + B2*ny = GB2(X,Y,T,H,Hx,Hy,E,Ex,Ey) [RETURN] and initial conditions: H = H0(X,Y) at T=T0 E = E0(X,Y) where H(X,Y,T) and E(X,Y,T) are the unknowns and C11,C12,F1,A1,B1, C21,C22,F2,A2,B2,FB1,FB2,GB1,GB2,H0,E0 are user-supplied functions. note: (nx,ny) = unit outward normal to the boundary Hx = d(H)/dX Ex = d(E)/dX Hy = d(H)/dY Ey = d(E)/dY Is this problem symmetric? If you don't want to read the FINE PRINT, it is safe to enter 'no'. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + This problem is called symmetric if each of the matrices + + + + F1.H F1.Hx F1.Hy F1.E F1.Ex F1.Ey + + A1.H A1.Hx A1.Hy A1.E A1.Ex A1.Ey + [RETURN] + B1.H B1.Hx B1.Hy B1.E B1.Ex B1.Ey + + F2.H F2.Hx F2.Hy F2.E F2.Ex F2.Ey + + A2.H A2.Hx A2.Hy A2.E A2.Ex A2.Ey + + B2.H B2.Hx B2.Hy B2.E B2.Ex B2.Ey + + + + C11 C12 + + C21 C22 + + and + + GB1.H GB1.E + + GB2.H GB2.E + + + + is always symmetric, where F1.H means d(F1)/d(H), and similarly + + for the other terms. In addition, GB1,GB2 must not depend on + + Hx,Hy,Ex,Ey. + + + + The memory and execution time are halved if the problem is known to + + be symmetric. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ If you don't want to read the FINE PRINT, enter 'no'. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ It can be verified with some effort that the PDE system is symmetric,$ [RETURN] $ and it will be seen later that GB1.E=GB2.H=0, so $ $ enter: yes $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no yes You may calculate one or more integrals (over the entire region) of some functions of the solution and its derivatives. How many integrals (NINT), if any, do you want to calculate? +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + In the FORTRAN program created by the preprocessor, the computed + + values of the integrals will be returned in the vector SINT8Z. If + + several iterations or time steps are done, only the last computed + + values are saved in SINT8Z (all values are printed). + + + + A limiting value, SLIM8Z(I), for the I-th integral can be set + + below in the main program. The computations will then stop + + gracefully whenever SINT8Z(I) > SLIM8Z(I), for any I=1...NINT. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: NINT = 1 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NINT = |---- Enter an integer value in the range 0 to 20 1 Enter FORTRAN expressions for the functions whose integrals are to be calculated and printed. They may be functions of X,Y,H,Hx,Hy,E,Ex,Ey and (if applicable) T The integrals may also contain references to the initial triangle number KTRI. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you only want to integrate a function over part of the region, + + define that function to be zero in the rest of the region. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ We will integrate abs(H)+abs(E), and identify eigenvalues as those $ $ values of beta (=T) which make this solution norm large. $ $ enter: INTEGRAL = ABS(H)+ABS(E) $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ INTEGRAL = |----Enter constant or FORTRAN expression-----------------------| ABS(H)+ABS(E) You may calculate one or more boundary integrals (over the entire boundary) of some functions of the solution and its derivatives. How many boundary integrals (NBINT), if any, do you want to calculate? +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + In the FORTRAN program created by the preprocessor, the computed + + values of the integrals will be returned in the vector BINT8Z. If + + several iterations or time steps are done, only the last computed + + values are saved in BINT8Z (all values are printed). + + + + A limiting value, BLIM8Z(I), for the I-th boundary integral can be + + set below in the main program. The computations will then stop + + gracefully whenever BINT8Z(I) > BLIM8Z(I), for any I=1...NBINT. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: NBINT = 0 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NBINT = |---- Enter an integer value in the range 0 to 20 0 If you don't want to read the FINE PRINT, enter 'no'. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Do you have point source terms in your PDEs, involving "Dirac Delta" + + functions? A Dirac Delta function DEL(x-xd0,y-yd0) is a function + + whose integral is 1, but which is zero everywhere except at a + + singular point (xd0,yd0). + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no Now enter FORTRAN expressions to define the PDE coefficients, which may be functions of X,Y,T,H,Hx,Hy,E,Ex,Ey They may also be functions of the initial triangle number KTRI. Recall that the PDEs have the form C11*d(H)/dT + C12*d(E)/dT = d/dX*A1 + d/dY*B1 - F1 C21*d(H)/dT + C22*d(E)/dT = d/dX*A2 + d/dY*B2 - F2 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ The partial differential equations may be written $ $ $ $ 0 = (E2)x + (-E1)y + w*amu*H + G1(X,Y) $ $ 0 = (-H2)x + (H1)y + w*eps*E + G2(X,Y) $ $ where $ $ E1 = (-w*amu*Hy - beta*Ex)/(w**2*amu*eps - beta**2) $ $ E2 = (w*amu*Hx - beta*Ey)/(w**2*amu*eps - beta**2) $ $ H1 = (w*eps*Ey - beta*Hx)/(w**2*amu*eps - beta**2) $ $ H2 = (-w*eps*Ex - beta*Hy)/(w**2*amu*eps - beta**2) $ [RETURN] $ G1,G2 = "random" nonhomogeneous terms $ $ $ $ When asked if you want to write a FORTRAN block, $ $ enter: yes $ $ then, when prompted, define: $ $ IF (ABS(X).LE.1.E-6 .AND. ABS(Y).LE.1.E-6) THEN $ $ C GLASS REGION $ $ EPS = 8.854 E-12*1.55**2 $ $ ELSE $ $ C SILICA REGION $ $ EPS = 8.854 E-12*1.50**2 $ $ ENDIF $ $ BETA = T $ $ DENOM = W**2*AMU*EPS - BETA**2 $ $ E1 = (-W*AMU*Hy - BETA*Ex)/DENOM $ $ E2 = ( W*AMU*Hx - BETA*Ey)/DENOM $ $ H1 = ( W*EPS*Ey - BETA*Hx)/DENOM $ $ H2 = (-W*EPS*Ex - BETA*Hy)/DENOM $ $ G1 = SIN(X/0.1E-6 + Y/0.2E-6) $ $ G2 = SIN(X/0.2E-6 + Y/0.1E-6) $ $ [blank line] $ $ then enter the following, when prompted: $ [RETURN] $ C11 = C12 = 0 F1 = -W*AMU*H - G1 A1 = E2 B1 = -E1 $ $ C21 = C22 = 0 F2 = -W*EPS*E - G2 A2 = -H2 B2 = H1 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ Do you want to write a FORTRAN block to define some parameters to be used in the definition of these coefficients? |---- Enter yes or no yes Remember to begin FORTRAN statements in column 7 |-----7-----Input FORTRAN now (type blank line to terminate)-----------| IF (ABS(X).LE.1.E-6 .AND. ABS(Y).LE.1.E-6) THEN C GLASS REGION EPS = 8.854 E-12*1.55**2 ELSE C SILICA REGION EPS = 8.854 E-12*1.50**2 ENDIF BETA = T DENOM = W**2*AMU*EPS - BETA**2 E1 = (-W*AMU*Hy - BETA*Ex)/DENOM E2 = ( W*AMU*Hx - BETA*Ey)/DENOM H1 = ( W*EPS*Ey - BETA*Hx)/DENOM H2 = (-W*EPS*Ex - BETA*Hy)/DENOM G1 = SIN(X/0.1E-6 + Y/0.2E-6) G2 = SIN(X/0.2E-6 + Y/0.1E-6) C(1,1) = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| 0 C(1,2) = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| 0 F1 = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| -W*AMU*H - G1 A1 = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| E2 B1 = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| -E1 C(2,1) = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| 0 C(2,2) = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| 0 F2 = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| -W*EPS*E - G2 A2 = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| -H2 B2 = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| H1 Now the initial values must be defined using FORTRAN expressions. They may be functions of X and Y, and of the initial triangle number KTRI. They may also reference the initial time T0. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ press [RETURN] to default H0 and E0 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ H0 = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| E0 = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| If you don't want to read the FINE PRINT, enter 'no'. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Do you want to read the initial conditions from the restart file, + + if it exists (and use the conditions supplied above if it does not + + exist)? + + + + If so, PDE2D will dump the final solution at the end of each run + + into a restart file "pde2d.res". Thus the usual procedure for + + using this dump/restart option is to make sure there is no restart + + file in your directory left over from a previous job, then the + + first time you run this job, the initial conditions supplied above + + will be used, but on the second and subsequent runs the restart file + + from the previous run will be used to define the initial conditions. + + + + You can do all the "runs" in one program, by setting NPROB > 1. + + Each pass through the DO loop, T0,TF,NSTEPS and possibly other + + parameters may be varied, by making them functions of IPROB. + + + + If the 2D or 3D collocation method is used, the coordinate + + transformation should not change between dump and restart. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ [RETURN] $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no Now the 'fixed' boundary conditions are described. Are there any boundary arcs with negative arc numbers, where nonzero 'fixed' boundary conditions must be specified? (The boundary conditions on arcs with negative numbers default to FB=0.) $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no Now the 'free' boundary conditions are described. Are there any boundary arcs with positive (but < 1000) arc numbers, where nonzero 'free' boundary conditions must be specified? (The boundary conditions on arcs with positive numbers default to GB=0.) $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: yes $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no yes Enter the arc number (IARC) of a boundary arc with positive arc number. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: IARC = 1 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ IARC = |---- Enter an integer value in the range 1 to 999 1 Enter FORTRAN expressions to define the following free boundary condition functions on this arc. They may be functions of X,Y,H,Hx,Hy,E,Ex,Ey and (if applicable) T +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + These functions may also reference the components (NORMx,NORMy) of + + the unit outward normal vector, the initial triangle number KTRI, + + and the arc parameter S (S = P or Q when INTRI=2). + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ Recall that free boundary conditions have the form A1*nx+B1*ny = GB1 A2*nx+B2*ny = GB2 +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Note that 'fixed' boundary conditions: + + Ui = FBi(X,Y,[T]) + + can be expressed as 'free' boundary conditions in the form: + + GBi = zero(Ui-FBi(X,Y,[T])) + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ [RETURN] $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ Recall that on the entire boundary we have dH/dn=E=0, and thus $ $ dE/ds=tangential derivative=0 also, so $ $ GB1 = A1*nx + B1*ny = E2*nx - E1*ny $ $ = (w*amu*dH/dn - beta*dE/ds)/(w**2*amu*eps-beta**2) = 0 $ $ GB2 = A2*nx + B2*ny = -H2*nx + H1*ny $ $ = (w*eps*dE/dn + beta*dH/ds)/(w**2*amu*eps-beta**2) = zero(E) $ $ The second boundary condition is (almost) equivalent to E=0. $ $ Thus enter, when prompted: $ $ GB1 = 0.0 $ $ GB2 = zero(E) $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ GB1 = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| 0.0 GB2 = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| zero(E) Are there any more positive arcs, with nonzero boundary conditions? $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no If you don't want to read the FINE PRINT, default all of the following variables. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Normally, PDE2D saves the values of H,A1,B1,E,A2,B2 at the + + output points. If different variables are to be saved (for later + + printing or plotting) the following functions can be used to + + re-define the output variables: + + define UPRINT(1) to replace H + + APRINT(1) A1 + + BPRINT(1) B1 + + UPRINT(2) E + + APRINT(2) A2 + + BPRINT(2) B2 + + Each function may be a function of + + + + X,Y,H,Hx,Hy,A1,B1,E,Ex,Ey,A2,B2 and (if applicable) T + + + + Each may also be a function of the initial triangle number KTRI and + + the integral estimates SINT(1),...,BINT(1),... + + + + The default for each variable is no change, for example, UPRINT(1) + [RETURN] + defaults to H. Enter FORTRAN expressions for each of the + + following functions (or default). + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ We want to save a norm of the solution in A1, and later plot this $ $ norm as a function of time, so $ $ enter: APRINT(1) = SINT(1) $ $ and default the other output modification variables $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ Replace H for postprocessing? UPRINT(1) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Replace A1 for postprocessing? APRINT(1) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| SINT(1) Replace B1 for postprocessing? BPRINT(1) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Replace E for postprocessing? UPRINT(2) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Replace A2 for postprocessing? APRINT(2) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Replace B2 for postprocessing? BPRINT(2) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| The solution is normally saved on a NX+1 by NY+1 rectangular grid of points (XA + I*(XB-XA)/NX , YA + J*(YB-YA)/NY) I=0,...,NX, J=0,...,NY. Enter values for NX and NY. Suggested values are NX=NY=25. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you want to save the solution at an arbitrary user-specified + + set of points, set NY=0 and NX+1=number of points. In this case you + + can request tabular output of the solution, but you cannot make any + + solution plots. + + + + If you set NEAR8Z=1 in the main program, the values saved at each + + output point will actually be the solution as evaluated at a nearby + + integration point. For most problems this obviously will produce + + less accurate output or plots, but for certain (rare) problems, a + + solution component may be much less noisy when plotted only at + + integration points. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: NX = 10 $ $ NY = 10 $ [RETURN] $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NX = |---- Enter an integer value in the range 1 to +INFINITY 10 NY = |---- Enter an integer value in the range 0 to +INFINITY 10 The solution is saved on an NX+1 by NY+1 rectangular grid covering the rectangle (XA,XB) x (YA,YB). Enter values for XA,XB,YA,YB. These variables are usually defaulted. The default is a rectangle which just covers the entire region. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ press [RETURN] to default XA,XB,YA,YB $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ XA = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| XB = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| YA = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| YB = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| The solution will be saved (for possible postprocessing) at the NSAVE+1 time points T0 + K*(TF-T0)/NSAVE K=0,...,NSAVE. Enter a value for NSAVE. If a user-specified constant time step is used, NSTEPS must be an integer multiple of NSAVE. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ We want to save the solution at all 101 values of T (=beta), so $ $ enter: NSAVE = 100 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NSAVE = |---- Enter an integer value in the range 1 to +INFINITY 100 To generate tabular or graphical output, choose an output option from the list below. 0. No further output is desired 1. Table of values at output points The tabulated output is saved in a file. 2. Surface plot of a scalar variable 3. Contour plot of a scalar variable 4. Vector field plot, with arrows indicating magnitude and direction 5. One-dimensional cross-sectional plots (versus X, Y or T) or, if applicable: 6. Stress field plot (requires 2 or more PDES) A plot of the principal stresses. Compression is indicated by arrows pointing toward each other, and tension by arrows pointing away from each other. Enter 0,1,2,3,4,5 or 6 to select an output option. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you set NY=0 and saved the solution at an arbitrary set of + + user-specified output points, you can only request tabular output. + + + [RETURN] + If you decide later that you want additional types of plots not + + requested during this interactive session, you will have to work + + through a new interactive session, so it is recommended that you + + request all output or plots you think you MIGHT eventually want now, + + during this session. + + + + Regardless of the options you select, a dummy subroutine POSTPR + + will be included in the program created by the interactive driver; + + you can add your own postprocessing code to this subroutine. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: 5, the first time you see this message and $ $ 0, the second time $ $ We could also (but won't) plot H and E for each value of beta (=T). $ $ Those corresponding to values of beta near eigenvalues will $ $ approximate the corresponding eigenfunctions; the others are not of $ $ interest. In addition, once the eigenvalues in the current range $ $ have been approximately located (by the peaks in the norm vs beta $ $ plot), other runs should probably be made with the eigenvalues $ $ bracketed more tightly, to yield more accurate estimates for the $ $ eigenvalues and eigenfunctions. $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ [RETURN] |---- Enter an integer value in the range 0 to 6 5 Enter a value for IVAR, to select the variable to be plotted or printed: IVAR = 1 means H (possibly as modified by UPRINT,..) 2 A1 3 B1 4 E 5 A2 6 B2 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ We want to plot the integral (solution norm), which has been saved $ $ in A1, as a function of beta (=T), so $ $ enter: IVAR = 2 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ IVAR = |---- Enter an integer value in the range 1 to 6 2 Which type of cross-sectional plots do you want? 1. Plots of output variable as function of X (constant Y [and T]) 2. Plots of output variable as function of Y (constant X [and T]) or, if applicable: 3. Plots of output variable as function of T (constant X and Y) Enter 1,2 or 3 to select a plot type. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: 3 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter an integer value in the range 1 to 3 3 One-dimensional plots of the output variable as a function of T will be made, at the output grid points (X,Y) closest to (XCROSS(I),YCROSS(J)), I=1,...,NXVALS, J=1,...,NYVALS Enter values for NXVALS, XCROSS(1),...,XCROSS(NXVALS), and NYVALS, YCROSS(1),...,YCROSS(NYVALS) $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ Since A1 = SINT(1) is independent of X and Y, all cross-sections $ $ will look the same, so $ $ enter: NXVALS = 1 $ $ XCROSS(1) = 0 $ $ NYVALS = 1 $ $ YCROSS(1) = 0 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NXVALS = |---- Enter an integer value in the range 1 to 100 1 XCROSS(1) = |----Enter constant or FORTRAN expression-----------------------| 0 NYVALS = |---- Enter an integer value in the range 1 to 100 1 YCROSS(1) = |----Enter constant or FORTRAN expression-----------------------| 0 Specify the range (UMIN,UMAX) for the dependent variable axis. UMIN and UMAX are often defaulted. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + By default, each plot will be scaled to just fit in the plot area. + + For a common scaling, you may want to set UMIN=ALOW, UMAX=AHIGH. + + ALOW and AHIGH are the minimum and maximum values over all output + + points and over all saved time steps or iterations. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ press [RETURN] to default UMIN and UMAX $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ UMIN = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| UMAX = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Enter a title, WITHOUT quotation marks. A maximum of 40 characters are allowed. The default is no title. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: Solution norm vs. beta $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ TITLE = (Press [RETURN] to default) |----Enter title or name---------------| Solution norm vs. beta To generate tabular or graphical output, choose an output option from the list below. 0. No further output is desired 1. Table of values at output points The tabulated output is saved in a file. 2. Surface plot of a scalar variable 3. Contour plot of a scalar variable 4. Vector field plot, with arrows indicating magnitude and direction 5. One-dimensional cross-sectional plots (versus X, Y or T) or, if applicable: 6. Stress field plot (requires 2 or more PDES) A plot of the principal stresses. Compression is indicated by arrows pointing toward each other, and tension by arrows pointing away from each other. Enter 0,1,2,3,4,5 or 6 to select an output option. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you set NY=0 and saved the solution at an arbitrary set of + + user-specified output points, you can only request tabular output. + + + [RETURN] + If you decide later that you want additional types of plots not + + requested during this interactive session, you will have to work + + through a new interactive session, so it is recommended that you + + request all output or plots you think you MIGHT eventually want now, + + during this session. + + + + Regardless of the options you select, a dummy subroutine POSTPR + + will be included in the program created by the interactive driver; + + you can add your own postprocessing code to this subroutine. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: 5, the first time you see this message and $ $ 0, the second time $ $ We could also (but won't) plot H and E for each value of beta (=T). $ $ Those corresponding to values of beta near eigenvalues will $ $ approximate the corresponding eigenfunctions; the others are not of $ $ interest. In addition, once the eigenvalues in the current range $ $ have been approximately located (by the peaks in the norm vs beta $ $ plot), other runs should probably be made with the eigenvalues $ $ bracketed more tightly, to yield more accurate estimates for the $ $ eigenvalues and eigenfunctions. $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ [RETURN] |---- Enter an integer value in the range 0 to 6 0 Do you want to define any FORTRAN function subprograms used in any of the FORTRAN 'expressions' entered earlier, entering them line by line? +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you selected double precision accuracy earlier, be sure to + + declare these functions and their arguments DOUBLE PRECISION. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no If you don't want to read the FINE PRINT, enter 'no'. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Do you want to define any FORTRAN function subprograms used in any + + of the FORTRAN 'expressions' entered earlier, by interpolating the + + tabular output saved in a file created on an earlier PDE2D run? + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 7 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no More detailed information about PDE2D can be found in the book "Solving Partial Differential Equation Applications with PDE2D," Granville Sewell, John Wiley & Sons, 2018. ******************************************* ***** Input program has been created ***** *******************************************