******************************************************* **** 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 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ In this example, the 2D axi-symmetric Navier-Stokes equations $ $ $ $ Visc*(Urr+Ur/r-U/r**2+Uzz) = Hr + Rho*(U*Ur+W*Uz) $ $ Visc*(Wrr+Wr/r +Wzz) = Hz + Rho*(U*Wr+W*Wz) $ $ Ur+Wz+U/r = 0 $ $ $ $ are solved in the channel -1 < z < 1, 0 < r < 1+z**2, where (U,W) $ $ are the radial and axial components of the fluid velocity, H is the $ $ pressure, and Visc=0.02 is the viscosity and Rho=1 is the density. $ $ $ $ We need to rename the variables r and z to X and Y, respectively. $ $ In this notation, the three equations can now be written: $ $ $ $ Visc*(Uxx+Ux/X-U/X**2+Uyy) - Hx - Rho*(U*Ux+W*Uy) = 0 $ $ Visc*(Wxx+Wx/X +Wyy) - Hy - Rho*(U*Wx+W*Wy) = 0 $ $ Ux+Wy+U/X = 0 $ $ $ $ Boundary conditions are $ $ $ $ 2*Visc*Wy-H=0, Uy+Wx=0 $ $ Y=1 ------------------------+ $ [RETURN] $ | + $ $ | + $ $ | + $ $ | + $ $ | + $ $ | + $ $ | + U=0 $ $ none | + W=0 $ $ | + $ $ | + $ $ | + $ $ | + $ $ | + $ $ | + $ $ | + $ $ Y=-1 ------------------------+ $ $ X=0 X=2 $ $ W = 1-(X/2)**2 $ $ Uy+Wx=0 $ $ H2=0 $ $ $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ [RETURN] [RETURN] In what follows, when you are told to enter a 'FORTRAN expression', this means any valid FORTRAN expression of 65 characters or less. In this expression, you may include references to FORTRAN function subprograms. You may define these functions line by line at the end of the interactive session, when prompted, or add them later using an editor. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you enter a "#" in the first column of any input line, this + + instructs the interactive driver to read this and subsequent input + + lines from the file "pde2d.in". A "#" in the first column of an + + input line in the file "pde2d.in" (or an end-of-file) instructs the + + driver to switch back to interactive input. + + + + All lines input during an interactive session are echo printed to + + a file "echo.out". You may want to modify this file and rename it + + "pde2d.in", and read some or all of your input from this file during + + your next interactive session. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ PDE2D can solve problems with 0,1,2 or 3 space variables. Enter the dimension of your problem: [RETURN] 0 - to solve a time-dependent ordinary differential equation system, or an algebraic or algebraic eigenvalue system 1 - to solve problems in 1D intervals 2 - to solve problems in general 2D regions 3 - to solve problems in a wide range of simple 3D regions $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: 2 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter an integer value in the range 0 to 3 2 Which finite element method do you want to use: 1. Galerkin method 2. Collocation method The collocation method can handle a wide range of simple 2D regions; the Galerkin method can handle completely general 2D regions. Enter 1 or 2 to select a finite element method. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you have a composite region, with discontinuous material + + parameters, you should use the Galerkin method. If your partial + + differential equations and boundary conditions are difficult to put + + into the "divergence" form required by the Galerkin method, or if + + you have periodic boundary conditions, use the collocation method. + + The collocation method produces an approximate solution with + + continuous first derivatives; the Galerkin solution is continuous + + but its first derivatives are not. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: 2 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ [RETURN] |---- Enter an integer value in the range 1 to 2 2 Is double precision mode to be used? Double precision is recommended. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If double precision mode is used, variables and functions assigned + + names beginning with a letter in the range A-H or O-Z will be DOUBLE + + PRECISION, and you should use double precision constants and FORTRAN + + expressions throughout; otherwise such variables and functions will + + be of type REAL. In either case, variables and functions assigned + + names beginning with I,J,K,L,M or N will be of INTEGER type. + + + + It is possible to convert a single precision PDE2D program to double + + precision after it has been created, using an editor. Just change + + all occurrences of "real" to "double precision" + + " tdp" to "dtdp" (note leading blank) + + Any user-written code or routines must be converted "by hand", of + + course. To convert from double to single, reverse the changes. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: yes $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no yes If you don't want to read the FINE PRINT, default NPROB. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you want to solve several similar problems in the same run, set + + NPROB equal to the number of problems you want to solve. Then NPROB + + loops through the main program will be done, with IPROB=1,...,NPROB, + + and you can make the problem parameters vary with IPROB. NPROB + + defaults to 1. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ press [RETURN] to default NPROB $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NPROB = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| PDE2D solves the time-dependent system (note: U,F,G,U0 may be vectors, C,RHO may be matrices): C(X,Y,T,U,Ux,Uy)*d(U)/dT = F(X,Y,T,U,Ux,Uy,Uxx,Uyy,Uxy) or the steady-state system: F(X,Y,U,Ux,Uy,Uxx,Uyy,Uxy) = 0 or the linear and homogeneous eigenvalue system: F(X,Y,U,Ux,Uy,Uxx,Uyy,Uxy) = lambda*RHO(X,Y)*U with boundary conditions: G(X,Y,[T],U,Ux,Uy) = 0 (periodic boundary conditions are also permitted) For time-dependent problems there are also initial conditions: U = U0(X,Y) at T=T0 [RETURN] +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If your PDEs involve the solution at points other than (P1,P2), the + + function + + (D)OLDSOL2(IDER,IEQ,PP1,PP2,KDEG) + + will interpolate (using interpolation of degree KDEG=1,2 or 3) to + + (PP1,PP2) 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 (PP1,PP2), assuming this has not + + been 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) = 0, where U = UR + UI*I + [RETURN] + would be difficult to split up analytically, but using FORTRAN + + expressions it is easy: + + F1 = -(UIxx+UIyy) - REAL(1.0/(1.0+CMPLX(UR,UI)**10)) + + F2 = (URxx+URyy) - 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. 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. [RETURN] +++++++++++++++ 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" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: VISC $ $ 0.02 $ $ RHO $ $ 1.0 $ $ [blank line] $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ Parameter name = (type blank line to terminate) VISC VISC = |----Enter constant or FORTRAN expression-----------------------| 0.02 Parameter name = (type blank line to terminate) RHO RHO = |----Enter constant or FORTRAN expression-----------------------| 1.0 Parameter name = (type blank line to terminate) If you don't want to read the FINE PRINT, enter 'no'. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Do you want to be given a chance to write a FORTRAN block before the + + definitions of many functions? If you answer 'no', you will still + + be given a chance to write code before the definition of the PDE + + coefficients, but not other functions. Of course, you can always + + add code later directly to the resulting program, using an editor. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no You can solve problems in your region only if you can describe it by X = X(P1,P2) Y = Y(P1,P2) with constant limits on the parameters P1,P2. If your region is rectangular, enter ITRANS=0 and the trivial parameterization X = P1 Y = P2 will be used. Otherwise, you need to read the FINE PRINT below. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If P1,P2 represent polar or other non-Cartesian coordinates, you can + + reference the Cartesian coordinates X,Y and derivatives of your + + unknowns with respect to these coordinates, when you define your + + PDE coefficients, boundary conditions, and volume and boundary + + integrals, if you enter ITRANS .NE. 0. Enter: + + ITRANS = 1, if P1,P2 are polar coordinates, that is, if + + P1=R, P2=Theta, where X = R*cos(Theta) + + Y = R*sin(Theta) + + ITRANS = -1, same as ITRANS=1, but P1=Theta, P2=R + + ITRANS = 3, to define your own coordinate transformation. In + + this case, you will be prompted to define X,Y and + + their first and second derivatives in terms of P1,P2. + [RETURN] + Because of symmetry, you will not be prompted for all + + of the second derivatives. If you make a mistake in + + computing any of these derivatives, PDE2D will usually + + be able to issue a warning message. (X1 = dX/dP1, etc) + + ITRANS = -3, same as ITRANS=3, but you will only be prompted to + + define X,Y; their first and second derivatives will + + be approximated using finite differences. + + When ITRANS = -3 or 3, the first derivatives of X,Y must all be + + continuous. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ Although in the X,Y (r,z) coordinate system the region is not $ $ rectangular, we can define a new coordinate system, with constant $ $ limits on the new variables: $ $ X = P1*(1+P2**2) 0 < P1 < 1 $ $ Y = P2 -1 < P2 < 1 $ $ so $ $ enter: ITRANS = -3 $ $ $ $ Then enter the following, when prompted: $ $ X = P1*(1+P2**2) $ $ Y = P2 $ [RETURN] $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ITRANS = |---- Enter an integer value in the range -3 to 3 -3 X = |----Enter constant or FORTRAN expression-----------------------| P1*(1+P2**2) Y = |----Enter constant or FORTRAN expression-----------------------| P2 A collocation finite element method is used, with bi-cubic Hermite basis functions on the elements (small rectangles) defined by the grid points: P1GRID(1),...,P1GRID(NP1GRID) P2GRID(1),...,P2GRID(NP2GRID) You will first be prompted for NP1GRID, the number of P1-grid points, then for P1GRID(1),...,P1GRID(NP1GRID). Any points defaulted will be uniformly spaced between the points you define; the first and last points cannot be defaulted. Then you will be prompted similarly for the number and values of the P2-grid points. The limits on the parameters are then: P1GRID(1) < P1 < P1GRID(NP1GRID) P2GRID(1) < P2 < P2GRID(NP2GRID) $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: NP1GRID = 20 $ $ P1GRID(1) = 0 $ $ P1GRID(NP1GRID) = 1 $ $ and default P1GRID(2),...,P1GRID(19) $ $ NP2GRID = 20 $ $ P2GRID(1) = -1 $ $ P2GRID(NP2GRID) = 1 $ [RETURN] $ and default P2GRID(2),...,P2GRID(19) $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NP1GRID = |---- Enter an integer value in the range 1 to +INFINITY 20 P1GRID(1) = |----Enter constant or FORTRAN expression-----------------------| 0 P1GRID(2) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(3) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(4) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(5) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(6) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(7) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(8) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(9) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(10) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(11) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(12) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(13) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(14) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(15) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(16) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(17) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(18) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(19) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P1GRID(NP1GRID) = |----Enter constant or FORTRAN expression-----------------------| 1 NP2GRID = |---- Enter an integer value in the range 1 to +INFINITY 20 P2GRID(1) = |----Enter constant or FORTRAN expression-----------------------| -1 P2GRID(2) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(3) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(4) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(5) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(6) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(7) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(8) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(9) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(10) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(11) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(12) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(13) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(14) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(15) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(16) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(17) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(18) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(19) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2GRID(NP2GRID) = |----Enter constant or FORTRAN expression-----------------------| 1 If you don't want to read the FINE PRINT, enter ISOLVE = 1. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + The following linear system solvers are available: + + + + 1. Sparse direct method + + Harwell Library routine MA27 (used by permission) is + + used to solve the (positive definite) "normal" + + equations A**T*A*x = A**T*b. The normal equations, + + which are essentially the equations which would result + + if a least squares finite element method were used + + instead of a collocation method, are substantially + + more ill-conditioned than the original system Ax = b, + + so it may be important to use high precision if this + + option is chosen. + + 2. Frontal method + + This is an out-of-core band solver. If you want to + + override the default number of rows in the buffer (11),+ + set a new value for NPMX8Z in the main program. + + 3. Jacobi conjugate gradient iterative method + + A preconditioned conjugate gradient iterative method + + is used to solve the (positive definite) normal + [RETURN] + equations. High precision is also important if this + + option is chosen. (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. Local solver (normal equations) + + 5. Local solver (original equations) + + Choose these options ONLY if alterative linear system + + 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. + [RETURN] ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ If you don't want to read the FINE PRINT, enter ISOLVE = 1. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: ISOLVE = 2 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ ISOLVE = |---- Enter an integer value in the range 1 to 6 2 What type of PDE problem do you want to solve? 1. a steady-state (time-independent) problem 2. a time-dependent problem 3. a linear, homogeneous eigenvalue problem Enter 1,2 or 3 to select a problem type. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: 1 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter an integer value in the range 1 to 3 1 Is this a linear problem? ("linear" means all differential equations and all boundary conditions are linear). If you aren't sure, it is safer to answer "no". $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no Give an upper limit on the number of Newton's method iterations (NSTEPS) to be allowed for this nonlinear problem. NSTEPS defaults to 15. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + The iteration will stop if convergence occurs before the upper + + limit has been reached. In the FORTRAN program created by the + + preprocessor, NCON8Z will be .TRUE. if Newton's method has converged.+ + + + For highly non-linear problems you may want to construct a one- + + parameter family of problems using the variable T, such that for + + T=1 the problems is easy (e.g. linear) and for T > N (N is less + + NSTEPS), the problem reduces to the original highly nonlinear + + problem. For example, the nonlinear term(s) may be multiplied by + + MIN(1.0,(T-1.0)/N). + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ press [RETURN] to default NSTEPS $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NSTEPS = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| How many differential equations (NEQN) are there in your problem? $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: NEQN = 3 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NEQN = |---- Enter an integer value in the range 1 to 99 3 You may now choose names for the component(s) of the (possibly vector) solution U. Each must be an alphanumeric string of one to three characters, beginning with a letter in the range A-H or O-Z. The variable names X,Y,P and T must not be used. The name should start in column 1. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: U1 = U $ $ U2 = W $ $ U3 = H $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ U1 = U U2 = W U3 = H You may calculate one or more integrals (over the entire region) of some functions of the solution and its derivatives. How many integrals (NINT), if any, do you want to calculate? +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + In the FORTRAN program created by the preprocessor, the computed + + values of the integrals will be returned in the vector SINT8Z. If + + several iterations or time steps are done, only the last computed + + values are saved in SINT8Z (all values are printed). + + + + A limiting value, SLIM8Z(I), for the I-th integral can be set + + below in the main program. The computations will then stop + + gracefully whenever SINT8Z(I) > SLIM8Z(I), for any I=1...NINT. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: NINT = 1 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NINT = |---- Enter an integer value in the range 0 to 20 1 Enter FORTRAN expressions for the functions whose integrals are to be calculated and printed. They may be functions of X,Y,U,Ux,Uy,Uxx,Uyy,Uxy W,Wx,Wy,Wxx,Wyy,Wxy H,Hx,Hy,Hxx,Hyy,Hxy and (if applicable) T The parameters P1,P2 and derivatives with respect to these may also be referenced (U1 = dU/dP1, etc): U1,U2,U11,U22,U12 W1,W2,W11,W22,W12 H1,H2,H11,H22,H12 +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you only want to integrate a function over part of the region, + + define that function to be zero in the rest of the region. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ We want to monitor the progress of the iteration by looking at the $ $ value of some integral of the solution, so $ $ enter: INTEGRAL = U+W $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ [RETURN] INTEGRAL = |----Enter constant or FORTRAN expression-----------------------| U+W You may calculate one or more boundary integrals (over the entire boundary) of some functions of the solution and its derivatives. How many boundary integrals (NBINT), if any, do you want to calculate? +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + In the FORTRAN program created by the preprocessor, the computed + + values of the integrals will be returned in the vector BINT8Z. If + + several iterations or time steps are done, only the last computed + + values are saved in BINT8Z (all values are printed). + + + + A limiting value, BLIM8Z(I), for the I-th boundary integral can be + + set below in the main program. The computations will then stop + + gracefully whenever BINT8Z(I) > BLIM8Z(I), for any I=1...NBINT. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: NBINT = 0 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ NBINT = |---- Enter an integer value in the range 0 to 20 0 Now enter FORTRAN expressions to define the PDE coefficients, which may be functions of X,Y,U,Ux,Uy,Uxx,Uyy,Uxy W,Wx,Wy,Wxx,Wyy,Wxy H,Hx,Hy,Hxx,Hyy,Hxy and, in some cases, of the parameter T. Recall that the PDEs have the form F1 = 0 F2 = 0 F3 = 0 The parameters P1,P2 and derivatives with respect to these may also be referenced (U1 = dU/dP1, etc): U1,U2,U11,U22,U12 W1,W2,W11,W22,W12 H1,H2,H11,H22,H12 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ The partial differential equations may be written $ [RETURN] $ $ $ Visc*(Uxx+Ux/X-U/X**2+Uyy) - Hx - Rho*(U*Ux+W*Uy) = 0 $ $ Visc*(Wxx+Wx/X +Wyy) - Hy - Rho*(U*Wx+W*Wy) = 0 $ $ Ux+Wy+U/X = 0 $ $ $ $ When asked if you want to write a FORTRAN block, $ $ enter: no $ $ then enter the following, when prompted: $ $ F1 = Visc*(Uxx+Ux/X-U/X**2+Uyy) - Hx - Rho*(U*Ux+W*Uy) $ $ F2 = Visc*(Wxx+Wx/X+Wyy) - Hy - Rho*(U*Wx+W*Wy) $ $ F3 = Ux+Wy+U/X $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ Do you want to write a FORTRAN block to define some parameters to be used in the definition of these coefficients? |---- Enter yes or no no F1 = |----Enter constant or FORTRAN expression-----------------------| Visc*(Uxx+Ux/X-U/X**2+Uyy) - Hx - Rho*(U*Ux+W*Uy) F2 = |----Enter constant or FORTRAN expression-----------------------| Visc*(Wxx+Wx/X+Wyy) - Hy - Rho*(U*Wx+W*Wy) F3 = |----Enter constant or FORTRAN expression-----------------------| Ux+Wy+U/X Now the initial values for Newton's method must be defined using FORTRAN expressions. They may be functions of X and Y (and the parameters P1,P2). It is important to provide initial values which are at least of the correct order of magnitude. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: U0 = 0 $ $ W0 = 1-(X/2)**2 $ $ H0 = 0 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ U0 = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| 0 W0 = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| 1-(X/2)**2 H0 = (Press [RETURN] to default to 0) |----Enter constant or FORTRAN expression-----------------------| 0 If you don't want to read the FINE PRINT, enter 'no'. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Do you want to read the initial conditions from the restart file, + + if it exists (and use the conditions supplied above if it does not + + exist)? + + + + If so, PDE2D will dump the final solution at the end of each run + + into a restart file "pde2d.res". Thus the usual procedure for + + using this dump/restart option is to make sure there is no restart + + file in your directory left over from a previous job, then the + + first time you run this job, the initial conditions supplied above + + will be used, but on the second and subsequent runs the restart file + + from the previous run will be used to define the initial conditions. + + + + You can do all the "runs" in one program, by setting NPROB > 1. + + Each pass through the DO loop, T0,TF,NSTEPS and possibly other + + parameters may be varied, by making them functions of IPROB. + + + + If the 2D or 3D collocation method is used, the coordinate + + transformation should not change between dump and restart. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ [RETURN] $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no If you do not have any periodic boundary conditions, enter IPERDC=0. Enter IPERDC=1 for periodic conditions at P1 = P1GRID(1),P1GRID(NP1GRID) IPERDC=2 for periodic conditions at P2 = P2GRID(1),P2GRID(NP2GRID) IPERDC=4 for periodic conditions on both P1 and P2 +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + When periodic boundary conditions are selected, they apply to all + + variables by default. To turn off periodic boundary conditions on + + the I-th variable, set PERDC(I) to 0 (or another appropriate value + + of IPERDC) below in the main program and set the desired boundary + + conditions in subroutine GB8Z, "by hand". + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: IPERDC = 0 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ IPERDC = |---- Enter an integer value in the range 0 to 4 0 Enter FORTRAN expressions to define the boundary condition functions, which may be functions of X,Y,U,Ux,Uy, W,Wx,Wy, H,Hx,Hy and (if applicable) T Recall that the boundary conditions have the form G1 = 0 G2 = 0 G3 = 0 Enter NONE to indicate "no" boundary condition. The parameters P1,P2 and derivatives with respect to these may also be referenced (U1 = dU/dP1, etc): U1,U2 W1,W2 H1,H2 The components (NORMx,NORMy) of the unit outward normal vector may also be referenced, as well as the normal derivatives Unorm, [RETURN] Wnorm,Hnorm. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If "no" boundary condition is specified, the corresponding PDE is + + enforced at points just inside the boundary (exactly on the + + boundary, if EPS8Z is set to 0 in the main program). + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ First define the boundary conditions on the face P1 = P1GRID(1). $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: G1 = NONE $ $ G2 = NONE $ $ G3 = NONE $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ G1 = |----Enter constant or FORTRAN expression-----------------------| NONE G2 = |----Enter constant or FORTRAN expression-----------------------| NONE G3 = |----Enter constant or FORTRAN expression-----------------------| NONE Now define the boundary conditions on the face P1 = P1GRID(NP1GRID). $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: G1 = U $ $ G2 = W $ $ G3 = NONE $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ G1 = |----Enter constant or FORTRAN expression-----------------------| U G2 = |----Enter constant or FORTRAN expression-----------------------| W G3 = |----Enter constant or FORTRAN expression-----------------------| NONE Now define the boundary conditions on the face P2 = P2GRID(1). $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: G1 = Uy+Wx $ $ G2 = W - (1-(X/2)**2) $ $ G3 = H2 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ G1 = |----Enter constant or FORTRAN expression-----------------------| Uy+Wx G2 = |----Enter constant or FORTRAN expression-----------------------| W - (1-(X/2)**2) G3 = |----Enter constant or FORTRAN expression-----------------------| H2 Now define the boundary conditions on the face P2 = P2GRID(NP2GRID). $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: G1 = Uy+Wx $ $ G2 = 2*Visc*Wy - H $ $ G3 = NONE $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ G1 = |----Enter constant or FORTRAN expression-----------------------| Uy+Wx G2 = |----Enter constant or FORTRAN expression-----------------------| 2*Visc*Wy - H G3 = |----Enter constant or FORTRAN expression-----------------------| NONE If you don't want to read the FINE PRINT, default all of the following variables. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Normally, PDE2D saves the values of U,Ux,Uy,W,Wx,Wy, + + H,Hx,Hy at the output points. If different variables are to + + be saved (for later printing or plotting) the following functions + + can be used to re-define the output variables: + + define UPRINT(1) to replace U + + UXPRINT(1) Ux + + UYPRINT(1) Uy + + UPRINT(2) W + + UXPRINT(2) Wx + + UYPRINT(2) Wy + + UPRINT(3) H + + UXPRINT(3) Hx + + UYPRINT(3) Hy + + Each function may be a function of + + + + X,Y,U,Ux,Uy,Uxx,Uyy,Uxy + + W,Wx,Wy,Wxx,Wyy,Wxy + + H,Hx,Hy,Hxx,Hyy,Hxy and (if applicable) T + [RETURN] + + + Each may also be a function of the integral estimates SINT(1),..., + + BINT(1),... + + + + The parameters P1,P2 and derivatives with respect to these may also + + be referenced (U1 = dU/dP1, etc): + + U1,U2,U11,U22,U12 + + W1,W2,W11,W22,W12 + + H1,H2,H11,H22,H12 + + + + The default for each variable is no change, for example, UPRINT(1) + + defaults to U. Enter FORTRAN expressions for each of the + + following functions (or default). + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ press [RETURN] to default all output modification variables $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ Replace U for postprocessing? UPRINT(1) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Replace Ux for postprocessing? UXPRINT(1) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Replace Uy for postprocessing? UYPRINT(1) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Replace W for postprocessing? UPRINT(2) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Replace Wx for postprocessing? UXPRINT(2) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Replace Wy for postprocessing? UYPRINT(2) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Replace H for postprocessing? UPRINT(3) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Replace Hx for postprocessing? UXPRINT(3) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Replace Hy for postprocessing? UYPRINT(3) = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| The solution is normally saved on an NP1+1 by NP2+1 rectangular grid of points, P1 = P1A + I*(P1B-P1A)/NP1, I = 0,...,NP1 P2 = P2A + J*(P2B-P2A)/NP2, J = 0,...,NP2 Enter values for NP1 and NP2. Suggested values: NP1=NP2=25. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you want to save the solution at an arbitrary user-specified set + + of points, set NP2=0 and NP1+1=number of points. In this case you + + can request tabular output, but no plots can be made. + + + + 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 + + collocation 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 + + collocation points. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: NP1 = 20 $ $ NP2 = 20 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ [RETURN] NP1 = |---- Enter an integer value in the range 1 to +INFINITY 20 NP2 = |---- Enter an integer value in the range 0 to +INFINITY 20 The solution is saved on an NP1+1 by NP2+1 rectangular grid covering the rectangle (P1A,P1B) x (P2A,P2B). Enter values for P1A,P1B,P2A,P2B. These variables are usually defaulted. The defaults are P1A = P1GRID(1), P1B = P1GRID(NP1GRID) P2A = P2GRID(1), P2B = P2GRID(NP2GRID) $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ To avoid division by 0 at r=0, $ $ enter: P1A = 0.001 $ $ and default the other variables $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ P1A = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| 0.001 P1B = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2A = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| P2B = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| To generate tabular or graphical output, choose an output option from the list below. 0. No further output is desired 1. Table of values at output points The tabulated output is saved in a file. 2. Surface plot of a scalar variable. Plots will only reflect the true geometry of the region if ITRANS=0. 3. Contour plot of a scalar variable. Plots will be made which reflect the true geometry of the region. 4. Vector field plot. Plots will be made which reflect the true geometry of the region. 5. One dimensional cross-sectional plots (versus P1, P2 or T) Enter 0,1,2,3,4 or 5 to select an output option. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you set NP2=0 and saved the solution at an arbitrary set of + + user-specified output points, you can only request tabular output. + + + + If you decide later that you want additional types of plots not + + requested during this interactive session, you will have to work + [RETURN] + through a new interactive session, so it is recommended that you + + request all output or plots you think you MIGHT eventually want now, + + during this session. + + + + Regardless of the options you select, a dummy subroutine POSTPR + + will be included in the program created by the interactive driver; + + you can add your own postprocessing code to this subroutine. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: 4, the first time you see this message and $ $ 3, the second time $ $ 0, the third time $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter an integer value in the range 0 to 5 4 Enter values for IVAR1, IVAR2 to select the components Vr1 and Vr2 of the vector to be plotted. IVAR1,IVAR2 = 1 means U (possibly as modified by UPRINT,...) 2 Ux 3 Uy 4 W 5 Wx 6 Wy 7 H 8 Hx 9 Hy Vr1 and Vr2 are assumed to be the components of the vector in Cartesian coordinates. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ We want a vector plot of (U,W), so $ $ enter: IVAR1 = 1 $ $ IVAR2 = 4 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ IVAR1 = |---- Enter an integer value in the range 1 to 9 1 IVAR2 = |---- Enter an integer value in the range 1 to 9 4 If you don't want to read the FINE PRINT, enter 'no'. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Do you want to scale the axes on the plot so that the region is + + undistorted? Otherwise the axes will be scaled so that the figure + + approximately fills the plot space. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: yes $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no yes For the purpose of scaling the arrows, the ranges of the two components of the vector are assumed to be (-VR1MAG,VR1MAG) and (-VR2MAG,VR2MAG). Enter values for VR1MAG and VR2MAG. VR1MAG and VR2MAG are often defaulted. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + By default, VR1MAG and VR2MAG are the maxima of the absolute values + + of the first and second components. For a common scaling, you may + + want to set VR1MAG=A1MAG, VR2MAG=A2MAG. A1MAG, A2MAG are the + + maxima of the absolute values over all output points and over all + + saved time steps or iterations. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ press [RETURN] to default VR1MAG and VR2MAG $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ VR1MAG = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| VR2MAG = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Enter a title, WITHOUT quotation marks. A maximum of 40 characters are allowed. The default is no title. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: Axisymmetric fluid flow problem $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ TITLE = (Press [RETURN] to default) |----Enter title or name---------------| Axisymmetric fluid flow problem To generate tabular or graphical output, choose an output option from the list below. 0. No further output is desired 1. Table of values at output points The tabulated output is saved in a file. 2. Surface plot of a scalar variable. Plots will only reflect the true geometry of the region if ITRANS=0. 3. Contour plot of a scalar variable. Plots will be made which reflect the true geometry of the region. 4. Vector field plot. Plots will be made which reflect the true geometry of the region. 5. One dimensional cross-sectional plots (versus P1, P2 or T) Enter 0,1,2,3,4 or 5 to select an output option. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you set NP2=0 and saved the solution at an arbitrary set of + + user-specified output points, you can only request tabular output. + + + + If you decide later that you want additional types of plots not + + requested during this interactive session, you will have to work + [RETURN] + through a new interactive session, so it is recommended that you + + request all output or plots you think you MIGHT eventually want now, + + during this session. + + + + Regardless of the options you select, a dummy subroutine POSTPR + + will be included in the program created by the interactive driver; + + you can add your own postprocessing code to this subroutine. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: 4, the first time you see this message and $ $ 3, the second time $ $ 0, the third time $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter an integer value in the range 0 to 5 3 Enter a value for IVAR, to select the variable to be plotted or printed: IVAR = 1 means U (possibly as modified by UPRINT,..) 2 Ux 3 Uy 4 W 5 Wx 6 Wy 7 H 8 Hx 9 Hy $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ We want a contour plot of H, so $ $ enter: IVAR = 7 $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ IVAR = |---- Enter an integer value in the range 1 to 9 7 If you don't want to read the FINE PRINT, enter 'no'. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Do you want to scale the axes on the plot so that the region is + + undistorted? Otherwise the axes will be scaled so that the figure + + approximately fills the plot space. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: yes $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no yes Enter lower (UMIN) and upper (UMAX) bounds for the contour values. UMIN and UMAX are often defaulted. Labeled contours will be drawn corresponding to the values UMIN + S*(UMAX-UMIN), for S=0.05,0.15,...0.95. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + By default, UMIN and UMAX are set to the minimum and maximum values + + of the variable to be plotted. For a common scaling, you may want + + to set UMIN=ALOW, UMAX=AHIGH. ALOW and AHIGH are the minimum and + + maximum values over all output points and over all saved time steps + + or iterations. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ press [RETURN] to default UMIN and UMAX $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ UMIN = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| UMAX = (Press [RETURN] to default) |----Enter constant or FORTRAN expression-----------------------| Do you want two additional unlabeled contours to be drawn between each pair of labeled contours? $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no Enter a title, WITHOUT quotation marks. A maximum of 40 characters are allowed. The default is no title. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: Axisymmetric fluid flow problem $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ TITLE = (Press [RETURN] to default) |----Enter title or name---------------| Axisymmetric fluid flow problem To generate tabular or graphical output, choose an output option from the list below. 0. No further output is desired 1. Table of values at output points The tabulated output is saved in a file. 2. Surface plot of a scalar variable. Plots will only reflect the true geometry of the region if ITRANS=0. 3. Contour plot of a scalar variable. Plots will be made which reflect the true geometry of the region. 4. Vector field plot. Plots will be made which reflect the true geometry of the region. 5. One dimensional cross-sectional plots (versus P1, P2 or T) Enter 0,1,2,3,4 or 5 to select an output option. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you set NP2=0 and saved the solution at an arbitrary set of + + user-specified output points, you can only request tabular output. + + + + If you decide later that you want additional types of plots not + + requested during this interactive session, you will have to work + [RETURN] + through a new interactive session, so it is recommended that you + + request all output or plots you think you MIGHT eventually want now, + + during this session. + + + + Regardless of the options you select, a dummy subroutine POSTPR + + will be included in the program created by the interactive driver; + + you can add your own postprocessing code to this subroutine. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: 4, the first time you see this message and $ $ 3, the second time $ $ 0, the third time $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter an integer value in the range 0 to 5 0 Do you want to define any FORTRAN function subprograms used in any of the FORTRAN 'expressions' entered earlier, entering them line by line? +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + If you selected double precision accuracy earlier, be sure to + + declare these functions and their arguments DOUBLE PRECISION. + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- Enter yes or no no If you don't want to read the FINE PRINT, enter 'no'. +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++ + Do you want to define any FORTRAN function subprograms used in any + + of the FORTRAN 'expressions' entered earlier, by interpolating the + + tabular output saved in a file created on an earlier PDE2D run? + ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ enter: no $ $$$$$$$$$$$$$$$$$$$$$$$$$$$$$ EXAMPLE 13 $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |---- 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 ***** *******************************************