C****************************** C Laplace Tidal Equations, with Bottom Drag (nonlinear) C C As discussed in "Solving the Laplace Tidal Equations using Freely C Available, Easily Extensible Finite Element Software," Sewell and C Manea, Computers and Geosciences, v155, October 2021. C C If run without changes, this program produces the result seen in line 1 C of Table 2 of that article and if 'mode' changed to 2, it produces C the result of line 1 of Table 3, but users can easily modify it to solve C many other tidal dissipation problems. Parameters most likely to C be modified by users are preceded by C??????????? C C Runs with the free PDE2D demo downloadable at www.pde2d.com C C The default output consists of the following plots (in PS file format): C U,V and ETA at 0%,25%,50%,75%,100% for the last period, and two plots C with the geographical distribution of tidal dissipation. In these C plots, P1=longitude, P2=latitude. C C To get MATLAB plots, remove all "C!" from LTE_nonlinear.f and pde2d.m and C pde2d.rdm will be produced when you run runpde2d. If you run MATLAB C program pde2d.m it will read data file pde2d.rdm and produce plots, C but way too many plots, replace pde2d.m by pde2d_LTE.m, available at C www.pde2d.com. C C****************************** C C ************************** C * PDE2D 9.6 MAIN PROGRAM * C ************************** C *** 2D PROBLEM SOLVED (COLLOCATION METHOD) *** C############################################################################## C Is double precision mode to be used? Double precision is recommended. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If double precision mode is used, variables and functions assigned +# C + names beginning with a letter in the range A-H or O-Z will be DOUBLE +# C + PRECISION, and you should use double precision constants and FORTRAN +# C + expressions throughout; otherwise such variables and functions will +# C + be of type REAL. In either case, variables and functions assigned +# C + names beginning with I,J,K,L,M or N will be of INTEGER type. +# C + +# C + It is possible to convert a single precision PDE2D program to double +# C + precision after it has been created, using an editor. Just change +# C + all occurrences of "real" to "double precision" +# C + " tdp" to "dtdp" (note leading blank) +# C + Any user-written code or routines must be converted "by hand", of +# C + course. To convert from double to single, reverse the changes. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## implicit double precision (a-h,o-z) parameter (neqnmx= 99) C############################################################################## C NP1GRID = number of P1-grid lines # C############################################################################## PARAMETER (NP1GRID = 41) C############################################################################## C NP2GRID = number of P2-grid lines # C############################################################################## PARAMETER (NP2GRID = 21) C############################################################################## C How many differential equations (NEQN) are there in your problem? # C############################################################################## PARAMETER (NEQN = 3) parameter (np3grid = 1) C DIMENSIONS OF WORK ARRAYS C SET TO 1 FOR AUTOMATIC ALLOCATION PARAMETER (IRWK8Z= 1) PARAMETER (IIWK8Z= 1) PARAMETER (NXP8Z=101,NYP8Z=101,KDEG8Z=1,NZP8Z=KDEG8Z+1) C############################################################################## C The solution is normally saved on an NP1+1 by NP2+1 rectangular grid # C of points, # C P1 = P1A + I*(P1B-P1A)/NP1, I = 0,...,NP1 # C P2 = P2A + J*(P2B-P2A)/NP2, J = 0,...,NP2 # C Enter values for NP1 and NP2. Suggested values: NP1=NP2=25. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If you want to save the solution at an arbitrary user-specified set +# C + of points, set NP2=0 and NP1+1=number of points. In this case you +# C + can request tabular output, but no plots can be made. +# C + +# C + If you set NEAR8Z=1 in the main program, the values saved at each +# C + output point will actually be the solution as evaluated at a nearby +# C + collocation point. For most problems this obviously will produce +# C + less accurate output or plots, but for certain (rare) problems, a +# C + solution component may be much less noisy when plotted only at +# C + collocation points. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## PARAMETER (NP1 = 90) PARAMETER (NP2 = 45) C############################################################################## C The solution will be saved (for possible postprocessing) at the NSAVE+1 # C time points # C T0 + K*(TF-T0)/NSAVE # C K=0,...,NSAVE. Enter a value for NSAVE. # C # C If a user-specified constant time step is used, NSTEPS must be an # C integer multiple of NSAVE. # C############################################################################## PARAMETER (NSAVE = 200) common/parm8z/ pi,Omega,Cd,sigma,RT,h0,alpha,e,theta0,g,mode, & rk2,rh2 dimension p1grid(np1grid),p2grid(np2grid),p3grid(np3grid),p1out8z( &0:np1,0:np2),p2out8z(0:np1,0:np2),p3out8z(0:np1,0:np2),p1cross(100 &),p2cross(100),tout8z(0:nsave) C dimension xres8z(nxp8z),yres8z(nyp8z),zres8z(nzp8z), C & ures8z(neqn,nxp8z,nyp8z,nzp8z) allocatable iwrk8z(:),rwrk8z(:) C dimension iwrk8z(iiwk8z),rwrk8z(irwk8z) character*40 title logical linear,crankn,noupdt,nodist,fillin,evcmpx,adapt,plot,lsqfi &t,fdiff,solid,econ8z,ncon8z,restrt,gridid common/dtdp14/ sint8z(20),bint8z(20),slim8z(20),blim8z(20) common/dtdp15/ evlr8z,ev0r,evli8z,ev0i,evcmpx common/dtdp16/ p8z,evr8z(50),evi8z(50) common/dtdp19/ toler(neqnmx),adapt common/dtdp30/ econ8z,ncon8z common/dtdp45/ perdc(neqnmx) common/dtdp46/ eps8z,cgtl8z,npmx8z,itype,near8z common/dtdp52/ nxa8z,nya8z,nza8z,kd8z common/dtdp53/ work8z(nxp8z*nyp8z*nzp8z+9) common/dtdp64/ amin8z(4*neqnmx),amax8z(4*neqnmx) common/dtdp76/ mdim8z,nx18z,ny18z,p1a,p1b,p2a,p2b,uout(0:np1,0:np2 &,4,neqn,0:nsave) pi = 4.0*atan(1.d0) zr8z = 0.0 nxa8z = nxp8z nya8z = nyp8z nza8z = nzp8z nx18z = np1+1 ny18z = np2+1 mdim8z = 4 kd8z = kdeg8z C############################################################################## C If you don't want to read the FINE PRINT, default NPROB. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If you want to solve several similar problems in the same run, set +# C + NPROB equal to the number of problems you want to solve. Then NPROB +# C + loops through the main program will be done, with IPROB=1,...,NPROB, +# C + and you can make the problem parameters vary with IPROB. NPROB +# C + defaults to 1. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C??????????? C Work through 10*NPROB periods. The first run, delete restart file C "pde2d.res" before starting. If you need to rerun (sometimes it takes C many periods for convergence) it will start from the end of the last C run, if you do not delete "pde2d.res". C NPROB = 2 do 78755 iprob=1,nprob C############################################################################## C PDE2D solves the time-dependent system (note: U,F,G,U0 may be vectors, # C C,RHO may be matrices): # C # C C(X,Y,T,U,Ux,Uy)*d(U)/dT = F(X,Y,T,U,Ux,Uy,Uxx,Uyy,Uxy) # C # C or the steady-state system: # C # C F(X,Y,U,Ux,Uy,Uxx,Uyy,Uxy) = 0 # C # C or the linear and homogeneous eigenvalue system: # C # C F(X,Y,U,Ux,Uy,Uxx,Uyy,Uxy) = lambda*RHO(X,Y)*U # C # C with boundary conditions: # C # C G(X,Y,[T],U,Ux,Uy) = 0 # C (periodic boundary conditions are also permitted) # C # C For time-dependent problems there are also initial conditions: # C # C U = U0(X,Y) at T=T0 # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If your PDEs involve the solution at points other than (P1,P2), the +# C + function +# C + (D)OLDSOL2(IDER,IEQ,PP1,PP2,KDEG) +# C + will interpolate (using interpolation of degree KDEG=1,2 or 3) to +# C + (PP1,PP2) the function saved in UOUT(*,*,IDER,IEQ,ISET) on the last +# C + time step or iteration (ISET) for which it has been saved. Thus, +# C + for example, if IDER=1, this will return the latest value of +# C + component IEQ of the solution at (PP1,PP2), assuming this has not +# C + been modified using UPRINT... If your equations involve integrals of +# C + the solution, for example, you can use (D)OLDSOL2 to approximate +# C + these using the solution from the last time step or iteration. +# C + +# C + CAUTION: For a steady-state or eigenvalue problem, you must reset +# C + NOUT=1 if you want to save the solution each iteration. +# C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++# C + A system of NEQN complex partial differential equations must be +# C + written as a system of 2*NEQN real equations, by separating the +# C + equations into their real and imaginary parts. However, note that +# C + the complex arithmetic abilities of FORTRAN can be used to simplify +# C + this separation. For example, the complex PDE: +# C + I*(Uxx+Uyy) - 1/(1+U**10) = 0, where U = UR + UI*I +# C + would be difficult to split up analytically, but using FORTRAN +# C + expressions it is easy: +# C + F1 = -(UIxx+UIyy) - REAL(1.0/(1.0+CMPLX(UR,UI)**10)) +# C + F2 = (URxx+URyy) - AIMAG(1.0/(1.0+CMPLX(UR,UI)**10)) +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C You may now define global parameters, which may be referenced in any # C of the "FORTRAN expressions" you input throughout the rest of this # C interactive session. You will be prompted alternately for parameter # C names and their values; enter a blank name when you are finished. # C # C Parameter names are valid FORTRAN variable names, starting in # C column 1. Thus each name consists of 1 to 6 alphanumeric characters, # C the first of which must be a letter. If the first letter is in the # C range I-N, the parameter must be an integer. # C # C Parameter values are either FORTRAN constants or FORTRAN expressions # C involving only constants and global parameters defined on earlier # C lines. They may also be functions of the problem number IPROB, if # C you are solving several similar problems in one run (NPROB > 1). Note # C that you are defining global CONSTANTS, not functions; e.g., parameter # C values may not reference any of the independent or dependent variables # C of your problem. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If you define other parameters here later, using an editor, you must +# C + add them to COMMON block /PARM8Z/ everywhere this block appears, if +# C + they are to be "global" parameters. +# C + +# C + The variable PI is already included as a global parameter, with an +# C + accurate value 3.14159... +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C??????????? C *** Titan data *** C revolution frequency (radians/sec) Omega = 0.455 D-5 PERIOD = 2.*pi/Omega C bottom drag coefficient Cd = 0.1 d0 C ocean density (kg/m^3) sigma = 1000.0 d0 C satellite radius (m) RT = 2.5747 D6 C ocean depth (m) h0 = 100. C eccentricity of orbit e = 0.0288D0 C obliquity theta0 = -0.32d0/180.*pi C surface graviational acceleration (m/sec^2) g = 1.35 d0 C mode = 1 for eccentricity tidal heating C mode = 2 for obliquity tidal heating mode = 1 C Love numbers rk2 = 0.120d0 rh2 = 0.221d0 C Rayleigh dissipation coefficient (1/sec) alpha = 0.0 C############################################################################## C A collocation finite element method is used, with bi-cubic Hermite # C basis functions on the elements (small rectangles) defined by the grid # C points: # C P1GRID(1),...,P1GRID(NP1GRID) # C P2GRID(1),...,P2GRID(NP2GRID) # C You will first be prompted for NP1GRID, the number of P1-grid points, # C then for P1GRID(1),...,P1GRID(NP1GRID). Any points defaulted will be # C uniformly spaced between the points you define; the first and last # C points cannot be defaulted. Then you will be prompted similarly # C for the number and values of the P2-grid points. The limits on the # C parameters are then: # C P1GRID(1) < P1 < P1GRID(NP1GRID) # C P2GRID(1) < P2 < P2GRID(NP2GRID) # C # C############################################################################## call dtdpwx(p1grid,np1grid,0) call dtdpwx(p2grid,np2grid,0) C P1GRID DEFINED P1GRID(1) = & 0 P1GRID(NP1GRID) = & 2*pi C P2GRID DEFINED P2GRID(1) = & -pi/2. P2GRID(NP2GRID) = & pi/2. C p3grid(1) = 0 call dtdpwx(p1grid,np1grid,1) call dtdpwx(p2grid,np2grid,1) C############################################################################## C If you don't want to read the FINE PRINT, enter ISOLVE = 1. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + The following linear system solvers are available: +# C + +# C + 1. Sparse direct method +# C + Harwell Library routine MA27 (used by permission) is +# C + used to solve the (positive definite) "normal" +# C + equations A**T*A*x = A**T*b. The normal equations, +# C + which are essentially the equations which would result +# C + if a least squares finite element method were used +# C + instead of a collocation method, are substantially +# C + more ill-conditioned than the original system Ax = b, +# C + so it may be important to use high precision if this +# C + option is chosen. +# C + 2. Frontal method +# C + This is an out-of-core band solver. If you want to +# C + override the default number of rows in the buffer (11),+# C + set a new value for NPMX8Z in the main program. +# C + 3. Jacobi conjugate gradient iterative method +# C + A preconditioned conjugate gradient iterative method +# C + is used to solve the (positive definite) normal +# C + equations. High precision is also important if this +# C + option is chosen. (This solver is MPI-enhanced, if +# C + MPI is available.) If you want to override the +# C + default convergence tolerance, set a new relative +# C + tolerance CGTL8Z in the main program. +# C + 4. Local solver (normal equations) +# C + 5. Local solver (original equations) +# C + Choose these options ONLY if alterative linear system +# C + solvers have been installed locally. See subroutines +# C + (D)TD3M, (D)TD3N in file (d)subs.f for instructions +# C + on how to add local solvers. +# C + 6. MPI-based parallel band solver +# C + This is a parallel solver which runs efficiently on +# C + multiple processor machines, under MPI. It is a +# C + band solver, with the matrix distributed over the +# C + available processors. Choose this option ONLY if the +# C + solver has been activated locally. See subroutine +# C + (D)TD3O in file (d)subs.f for instructions on how to +# C + activate this solver and the MPI-enhancements to the +# C + conjugate gradient solver. +# C + +# C + Enter ISOLVE = 1,2,3,4,5 or 6 to select a linear system solver. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C # C If you don't want to read the FINE PRINT, enter ISOLVE = 1. # C############################################################################## ISOLVE = 1 C *******TIME-DEPENDENT PROBLEM itype = 2 C############################################################################## C Enter the initial time value (T0) and the final time value (TF), for # C this time-dependent problem. T0 defaults to 0. # C # C TF is not required to be greater than T0. # C############################################################################## T0 = 0.0 T0 = & 0 TF = & 10*PERIOD C############################################################################## C Is this a linear problem? ("linear" means all differential equations # C and all boundary conditions are linear). If you aren't sure, it is # C safer to answer "no". # C############################################################################## LINEAR = .FALSE. C############################################################################## C Do you want the time step to be chosen adaptively? If you answer # C 'yes', you will then be prompted to enter a value for TOLER(1), the # C local relative time discretization error tolerance. The default is # C TOLER(1)=0.01. If you answer 'no', a user-specified constant time step # C will be used. We suggest that you answer 'yes' and default TOLER(1) # C (although for certain linear problems, a constant time step may be much # C more efficient). # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If a negative value is specified for TOLER(1), then ABS(TOLER(1)) is +# C + taken to be the "absolute" error tolerance. If a system of PDEs is +# C + solved, by default the error tolerance specified in TOLER(1) applies +# C + to all variables, but the error tolerance for the J-th variable can +# C + be set individually by specifying a value for TOLER(J) using an +# C + editor, after the end of the interactive session. +# C + +# C + Each time step, two steps of size dt/2 are taken, and that solution +# C + is compared with the result when one step of size dt is taken. If +# C + the maximum difference between the two answers is less than the +# C + tolerance (for each variable), the time step dt is accepted (and the +# C + next step dt is doubled, if the agreement is "too" good); otherwise +# C + dt is halved and the process is repeated. Note that forcing the +# C + local (one-step) error to be less than the tolerance does not +# C + guarantee that the global (cumulative) error is less than that value.+# C + However, as the tolerance is decreased, the global error should +# C + decrease correspondingly. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## ADAPT = .TRUE. if (iprob.eq.1) then TOLER(1) = 1.D20 else TOLER(1) = 0.05D0 endif NOUPDT = .FALSE. C############################################################################## C The time stepsize will be chosen adaptively, between an upper limit # C of DTMAX = (TF-T0)/NSTEPS and a lower limit of 0.0001*DTMAX. Enter # C a value for NSTEPS (the minimum number of steps). # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If you later turn off adaptive time step control, the time stepsize +# C + will be constant, DT = (TF-T0)/NSTEPS. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C??????????? C MINIMUM number of time steps per 10-period run (must be multiple of 200) NSTEPS = & 400 dt = (tf-t0)/max(nsteps,1) C############################################################################## C If you don't want to read the FINE PRINT, enter 'no'. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + Is the Crank-Nicolson scheme to be used to discretize time? If you +# C + answer 'no', a backward Euler scheme will be used. +# C + +# C + If a user-specified constant time step is chosen, the second order +# C + Crank Nicolson method is recommended only for problems with very +# C + well-behaved solutions, and the first order backward Euler scheme +# C + should be used for more difficult problems. In particular, do not +# C + use the Crank Nicolson method if the left hand side of any PDE is +# C + zero, for example, if a mixed elliptic/parabolic problem is solved. +# C + +# C + If adaptive time step control is chosen, however, an extrapolation +# C + is done between the 1-step and 2-step answers which makes the Euler +# C + method second order, and the Crank-Nicolson method strongly stable. +# C + Thus in this case, both methods have second order accuracy, and both +# C + are strongly stable. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## CRANKN = .FALSE. C############################################################################## C If you don't want to read the FINE PRINT, enter 'yes' (strongly # C recommended). # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + The partial derivatives of some of the PDE and boundary condition +# C + coefficients are required by PDE2D. These may be calculated +# C + automatically using a finite difference approximation, or supplied +# C + by the user. Do you want them to be calculated automatically? +# C + +# C + If you answer 'yes', you will not be asked to supply the derivatives,+# C + but there is a small risk that the inaccuracies introduced by the +# C + finite difference approximation may cause the Newton iteration +# C + to converge more slowly or to diverge, especially if low precision +# C + is used. This risk is very low, however, and since answering 'no' +# C + means you may have to compute many partial derivatives, it is +# C + recommended you answer 'yes' unless you have some reason to believe +# C + there is a problem with the finite difference approximations. +# C + +# C + If you supply analytic partial derivatives, PDE2D will do some spot +# C + checking and can usually issue a warning if any are supplied +# C + incorrectly. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## FDIFF = .TRUE. C############################################################################## C You may calculate one or more integrals (over the entire region) of # C some functions of the solution and its derivatives. How many integrals # C (NINT), if any, do you want to calculate? # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + In the FORTRAN program created by the preprocessor, the computed +# C + values of the integrals will be returned in the vector SINT8Z. If +# C + several iterations or time steps are done, only the last computed +# C + values are saved in SINT8Z (all values are printed). +# C + +# C + A limiting value, SLIM8Z(I), for the I-th integral can be set +# C + below in the main program. The computations will then stop +# C + gracefully whenever SINT8Z(I) > SLIM8Z(I), for any I=1...NINT. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## NINT = 1 C############################################################################## C You may calculate one or more boundary integrals (over the entire # C boundary) of some functions of the solution and its derivatives. How # C many boundary integrals (NBINT), if any, do you want to calculate? # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + In the FORTRAN program created by the preprocessor, the computed +# C + values of the integrals will be returned in the vector BINT8Z. If +# C + several iterations or time steps are done, only the last computed +# C + values are saved in BINT8Z (all values are printed). +# C + +# C + A limiting value, BLIM8Z(I), for the I-th boundary integral can be +# C + set below in the main program. The computations will then stop +# C + gracefully whenever BINT8Z(I) > BLIM8Z(I), for any I=1...NBINT. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## NBINT = 0 C############################################################################## C If you don't want to read the FINE PRINT, enter 'no'. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + Normally, interpolation is done to approximate the initial values +# C + using cubic Hermites. Since some derivatives must be interpolated, +# C + if the initial values are not smooth (ie, have large or infinite +# C + derivatives), the resulting cubic interpolants may have undesired +# C + noise or large spikes. Do you want to compute a least squares +# C + approximation to the initial values, rather than an interpolant? +# C + The least squares fit is generally much smoother, but requires one +# C + extra linear system solution. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## LSQFIT = .FALSE. C############################################################################## C If you don't want to read the FINE PRINT, enter 'no'. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + Do you want to read the initial conditions from the restart file, +# C + if it exists (and use the conditions supplied above if it does not +# C + exist)? +# C + +# C + If so, PDE2D will dump the final solution at the end of each run +# C + into a restart file "pde2d.res". Thus the usual procedure for +# C + using this dump/restart option is to make sure there is no restart +# C + file in your directory left over from a previous job, then the +# C + first time you run this job, the initial conditions supplied above +# C + will be used, but on the second and subsequent runs the restart file +# C + from the previous run will be used to define the initial conditions. +# C + +# C + You can do all the "runs" in one program, by setting NPROB > 1. +# C + Each pass through the DO loop, T0,TF,NSTEPS and possibly other +# C + parameters may be varied, by making them functions of IPROB. +# C + +# C + If the 2D or 3D collocation method is used, the coordinate +# C + transformation should not change between dump and restart. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## RESTRT = .TRUE. C GRIDID = .FALSE. IF FINITE ELEMENT GRID CHANGES BETWEEN DUMP, RESTART GRIDID = .TRUE. C############################################################################## C If you do not have any periodic boundary conditions, enter IPERDC=0. # C # C Enter IPERDC=1 for periodic conditions at P1 = P1GRID(1),P1GRID(NP1GRID)# C IPERDC=2 for periodic conditions at P2 = P2GRID(1),P2GRID(NP2GRID)# C IPERDC=4 for periodic conditions on both P1 and P2 # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + When periodic boundary conditions are selected, they apply to all +# C + variables by default. To turn off periodic boundary conditions on +# C + the I-th variable, set PERDC(I) to 0 (or another appropriate value +# C + of IPERDC) below in the main program and set the desired boundary +# C + conditions in subroutine GB8Z, "by hand". +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## IPERDC = 1 C############################################################################## C The solution is saved on an NP1+1 by NP2+1 rectangular grid covering # C the rectangle (P1A,P1B) x (P2A,P2B). Enter values for P1A,P1B,P2A,P2B. # C These variables are usually defaulted. # C # C The defaults are P1A = P1GRID(1), P1B = P1GRID(NP1GRID) # C P2A = P2GRID(1), P2B = P2GRID(NP2GRID) # C # C############################################################################## C defaults for p1a,p1b,p2a,p2b p1a = p1grid(1) p1b = p1grid(np1grid) p2a = p2grid(1) p2b = p2grid(np2grid) C DEFINE P1A,P1B,P2A,P2B IMMEDIATELY BELOW: call dtdpx3(np1,np2,0,p1a,p1b,p2a,p2b,zr8z,zr8z,hp18z,hp28z,hp38z, &p1out8z,p2out8z,p3out8z,npts8z) call dtdpqx(np1grid,np2grid,np3grid,isolve,neqn,ii8z,ir8z,iperdc) if (iiwk8z.gt.1) ii8z = iiwk8z if (irwk8z.gt.1) ir8z = irwk8z C *******allocate workspace allocate (iwrk8z(ii8z),rwrk8z(ir8z)) C *******DRAW GRID LINES? PLOT = .FALSE. C *******call pde solver call dtdp3x(p1grid, p2grid, p3grid, np1grid,np2grid, -1, neqn, p1o &ut8z, p2out8z, p3out8z, uout, tout8z, npts8z, t0, dt, nsteps, nout &, nsave, crankn, noupdt, itype, linear, isolve, rwrk8z, ir8z, iwrk &8z, ii8z, iperdc, plot, lsqfit, fdiff, nint, nbint, restrt, gridid &) deallocate (iwrk8z,rwrk8z) C *******read from restart file to array ures8z C call dtdpr3(1,xres8z,nxp8z,yres8z,nyp8z,zres8z,nzp8z,ures8z,neqn) C *******write array ures8z back to restart file C call dtdpr3(2,xres8z,nxp8z,yres8z,nyp8z,zres8z,nzp8z,ures8z,neqn) C *******call user-written postprocessor call postpr(tout8z,nsave,p1out8z,p2out8z,np1,np2,uout,neqn, & iprob,nprob) if (iprob.lt.nprob) go to 78755 C *******SURFACE PLOTS C############################################################################## C Enter a value for IVAR, to select the variable to be plotted or # C printed: # C IVAR = 1 means U (possibly as modified by UPRINT,..) # C 2 Ux # C 3 Uy # C 4 V # C 5 Vx # C 6 Vy # C 7 ETA # C 8 ETAx # C 9 ETAy # C############################################################################## IVAR = 1 ivara8z = mod(ivar-1,3)+1 ivarb8z = (ivar-1)/3+1 C############################################################################## C If you don't want to read the FINE PRINT, default ISET1,ISET2,ISINC. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + The tabular output or plots will be made at times: +# C + T(K) = T0 + K*(TF-T0)/NSAVE +# C + for K = ISET1, ISET1+ISINC, ISET1+2*ISINC,..., ISET2 +# C + Enter values for ISET1, ISET2 and ISINC. +# C + +# C + The default is ISET1=0, ISET2=NSAVE, ISINC=1, that is, the tabular +# C + output or plots will be made at all time values for which the +# C + solution has been saved. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## ISET1 = 180 ISET2 = 200 ISINC = 5 C############################################################################## C Enter the view latitude, VLAT, and the view longitude, VLON, desired # C for this plot, in degrees. VLAT and VLON must be between 10 and 80 # C degrees; each defaults to 45 degrees. VLAT and VLON are usually # C defaulted. # C############################################################################## VLON = 45.0 VLAT = 45.0 C ivar8z = 4*(ivarb8z-1)+ivara8z alow = amin8z(ivar8z) ahigh = amax8z(ivar8z) C############################################################################## C Specify the range (UMIN,UMAX) for the dependent variable axis. UMIN # C and UMAX are often defaulted. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + By default, each plot will be scaled to just fit in the plot area. +# C + For a common scaling, you may want to set UMIN=ALOW, UMAX=AHIGH. +# C + ALOW and AHIGH are the minimum and maximum values over all output +# C + points and over all saved time steps or iterations. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## UMIN = alow UMAX = ahigh C############################################################################## C Enter a title, WITHOUT quotation marks. A maximum of 40 characters # C are allowed. The default is no title. # C############################################################################## TITLE = ' ' TITLE = 'U (m/sec) ' call dtdprx(tout8z,nsave,iset1,iset2,isinc) do 78757 is8z=iset1,iset2,isinc call dtdplo(p1out8z,p2out8z,p3out8z,uout(0,0,ivara8z,ivarb8z,is8z) &,np1,np2,0,3,ix8z,jy8z,0,title,vlon,vlat,umin,umax,tout8z(is8z)) 78757 continue C *******SURFACE PLOTS C############################################################################## C Enter a value for IVAR, to select the variable to be plotted or # C printed: # C IVAR = 1 means U (possibly as modified by UPRINT,..) # C 2 Ux # C 3 Uy # C 4 V # C 5 Vx # C 6 Vy # C 7 ETA # C 8 ETAx # C 9 ETAy # C############################################################################## IVAR = 4 ivara8z = mod(ivar-1,3)+1 ivarb8z = (ivar-1)/3+1 C############################################################################## C If you don't want to read the FINE PRINT, default ISET1,ISET2,ISINC. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + The tabular output or plots will be made at times: +# C + T(K) = T0 + K*(TF-T0)/NSAVE +# C + for K = ISET1, ISET1+ISINC, ISET1+2*ISINC,..., ISET2 +# C + Enter values for ISET1, ISET2 and ISINC. +# C + +# C + The default is ISET1=0, ISET2=NSAVE, ISINC=1, that is, the tabular +# C + output or plots will be made at all time values for which the +# C + solution has been saved. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## ISET1 = 180 ISET2 = 200 ISINC = 5 C############################################################################## C Enter the view latitude, VLAT, and the view longitude, VLON, desired # C for this plot, in degrees. VLAT and VLON must be between 10 and 80 # C degrees; each defaults to 45 degrees. VLAT and VLON are usually # C defaulted. # C############################################################################## VLON = 45.0 VLAT = 45.0 C ivar8z = 4*(ivarb8z-1)+ivara8z alow = amin8z(ivar8z) ahigh = amax8z(ivar8z) C############################################################################## C Specify the range (UMIN,UMAX) for the dependent variable axis. UMIN # C and UMAX are often defaulted. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + By default, each plot will be scaled to just fit in the plot area. +# C + For a common scaling, you may want to set UMIN=ALOW, UMAX=AHIGH. +# C + ALOW and AHIGH are the minimum and maximum values over all output +# C + points and over all saved time steps or iterations. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## UMIN = alow UMAX = ahigh C############################################################################## C Enter a title, WITHOUT quotation marks. A maximum of 40 characters # C are allowed. The default is no title. # C############################################################################## TITLE = ' ' TITLE = 'V (m/sec) ' call dtdprx(tout8z,nsave,iset1,iset2,isinc) do 78758 is8z=iset1,iset2,isinc call dtdplo(p1out8z,p2out8z,p3out8z,uout(0,0,ivara8z,ivarb8z,is8z) &,np1,np2,0,3,ix8z,jy8z,0,title,vlon,vlat,umin,umax,tout8z(is8z)) 78758 continue C *******SURFACE PLOTS C############################################################################## C Enter a value for IVAR, to select the variable to be plotted or # C printed: # C IVAR = 1 means U (possibly as modified by UPRINT,..) # C 2 Ux # C 3 Uy # C 4 V # C 5 Vx # C 6 Vy # C 7 ETA # C 8 ETAx # C 9 ETAy # C############################################################################## IVAR = 7 ivara8z = mod(ivar-1,3)+1 ivarb8z = (ivar-1)/3+1 C############################################################################## C If you don't want to read the FINE PRINT, default ISET1,ISET2,ISINC. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + The tabular output or plots will be made at times: +# C + T(K) = T0 + K*(TF-T0)/NSAVE +# C + for K = ISET1, ISET1+ISINC, ISET1+2*ISINC,..., ISET2 +# C + Enter values for ISET1, ISET2 and ISINC. +# C + +# C + The default is ISET1=0, ISET2=NSAVE, ISINC=1, that is, the tabular +# C + output or plots will be made at all time values for which the +# C + solution has been saved. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## ISET1 = 180 ISET2 = 200 ISINC = 5 C############################################################################## C Enter the view latitude, VLAT, and the view longitude, VLON, desired # C for this plot, in degrees. VLAT and VLON must be between 10 and 80 # C degrees; each defaults to 45 degrees. VLAT and VLON are usually # C defaulted. # C############################################################################## VLON = 45.0 VLAT = 45.0 C ivar8z = 4*(ivarb8z-1)+ivara8z alow = amin8z(ivar8z) ahigh = amax8z(ivar8z) C############################################################################## C Specify the range (UMIN,UMAX) for the dependent variable axis. UMIN # C and UMAX are often defaulted. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + By default, each plot will be scaled to just fit in the plot area. +# C + For a common scaling, you may want to set UMIN=ALOW, UMAX=AHIGH. +# C + ALOW and AHIGH are the minimum and maximum values over all output +# C + points and over all saved time steps or iterations. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## UMIN = alow UMAX = ahigh C############################################################################## C Enter a title, WITHOUT quotation marks. A maximum of 40 characters # C are allowed. The default is no title. # C############################################################################## TITLE = ' ' TITLE = 'ETA (m) ' call dtdprx(tout8z,nsave,iset1,iset2,isinc) do 78759 is8z=iset1,iset2,isinc call dtdplo(p1out8z,p2out8z,p3out8z,uout(0,0,ivara8z,ivarb8z,is8z) &,np1,np2,0,3,ix8z,jy8z,0,title,vlon,vlat,umin,umax,tout8z(is8z)) 78759 continue C *******SURFACE PLOTS C############################################################################## C Enter a value for IVAR, to select the variable to be plotted or # C printed: # C IVAR = 1 means U (possibly as modified by UPRINT,..) # C 2 Ux # C 3 Uy # C 4 V # C 5 Vx # C 6 Vy # C 7 ETA # C 8 ETAx # C 9 ETAy # C############################################################################## IVAR = 6 ivara8z = mod(ivar-1,3)+1 ivarb8z = (ivar-1)/3+1 C############################################################################## C If you don't want to read the FINE PRINT, default ISET1,ISET2,ISINC. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + The tabular output or plots will be made at times: +# C + T(K) = T0 + K*(TF-T0)/NSAVE +# C + for K = ISET1, ISET1+ISINC, ISET1+2*ISINC,..., ISET2 +# C + Enter values for ISET1, ISET2 and ISINC. +# C + +# C + The default is ISET1=0, ISET2=NSAVE, ISINC=1, that is, the tabular +# C + output or plots will be made at all time values for which the +# C + solution has been saved. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## ISET1 = NSAVE ISET2 = NSAVE ISINC = 1 C############################################################################## C Enter the view latitude, VLAT, and the view longitude, VLON, desired # C for this plot, in degrees. VLAT and VLON must be between 10 and 80 # C degrees; each defaults to 45 degrees. VLAT and VLON are usually # C defaulted. # C############################################################################## VLON = 45.0 VLAT = 45.0 C ivar8z = 4*(ivarb8z-1)+ivara8z alow = amin8z(ivar8z) ahigh = amax8z(ivar8z) C############################################################################## C Specify the range (UMIN,UMAX) for the dependent variable axis. UMIN # C and UMAX are often defaulted. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + By default, each plot will be scaled to just fit in the plot area. +# C + For a common scaling, you may want to set UMIN=ALOW, UMAX=AHIGH. +# C + ALOW and AHIGH are the minimum and maximum values over all output +# C + points and over all saved time steps or iterations. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## UMIN = 0 UMAX = 0 C############################################################################## C Enter a title, WITHOUT quotation marks. A maximum of 40 characters # C are allowed. The default is no title. # C############################################################################## TITLE = ' ' TITLE = 'Energy dissipation rate (W/m^2)' call dtdprx(tout8z,nsave,iset1,iset2,isinc) do 78769 is8z=iset1,iset2,isinc call dtdplo(p1out8z,p2out8z,p3out8z,uout(0,0,ivara8z,ivarb8z,is8z) &,np1,np2,0,3,ix8z,jy8z,0,title,vlon,vlat,umin,umax,tout8z(is8z)) 78769 continue C *******CONTOUR PLOTS C############################################################################## C Enter a value for IVAR, to select the variable to be plotted or # C printed: # C IVAR = 1 means U (possibly as modified by UPRINT,..) # C 2 Ux # C 3 Uy # C 4 W # C 5 Wx # C 6 Wy # C 7 H # C 8 Hx # C 9 Hy # C############################################################################## IVAR = 6 ivara8z = mod(ivar-1,3)+1 ivarb8z = (ivar-1)/3+1 ISET1 = NSAVE ISET2 = NSAVE ISINC = 1 C############################################################################## C If you don't want to read the FINE PRINT, enter 'no'. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + Do you want to scale the axes on the plot so that the region is +# C + undistorted? Otherwise the axes will be scaled so that the figure +# C + approximately fills the plot space. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## NODIST = .FALSE. C ivar8z = 4*(ivarb8z-1)+ivara8z alow = amin8z(ivar8z) ahigh = amax8z(ivar8z) C############################################################################## C Enter lower (UMIN) and upper (UMAX) bounds for the contour values. UMIN # C and UMAX are often defaulted. # C # C Labeled contours will be drawn corresponding to the values # C # C UMIN + S*(UMAX-UMIN), for S=0.05,0.15,...0.95. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + By default, UMIN and UMAX are set to the minimum and maximum values +# C + of the variable to be plotted. For a common scaling, you may want +# C + to set UMIN=ALOW, UMAX=AHIGH. ALOW and AHIGH are the minimum and +# C + maximum values over all output points and over all saved time steps +# C + or iterations. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## UMIN = 0.0 UMAX = 0.0 C############################################################################## C Do you want two additional unlabeled contours to be drawn between each # C pair of labeled contours? # C############################################################################## FILLIN = .FALSE. C############################################################################## C Enter a title, WITHOUT quotation marks. A maximum of 40 characters # C are allowed. The default is no title. # C############################################################################## TITLE = ' ' TITLE = 'Energy dissipation rate (W/m^2)' call dtdprx(tout8z,nsave,iset1,iset2,isinc) do 78760 is8z=iset1,iset2,isinc call dtdpln(uout(0,0,ivara8z,ivarb8z,is8z),np1,np2,0,p1a,p1b,p2a,p &2b,zr8z,zr8z,3,ix8z,jy8z,0,title,umin,umax,nodist,fillin,tout8z(is &8z),zr8z,zr8z,zr8z,zr8z,2,ical8z) 78760 continue 78755 continue call endgks stop end subroutine tran8z(itrans,p1,p2,p38z) implicit double precision (a-h,o-z) common /dtdp41/x,y,z8z,x1,x2,x38z,y1,y2,y38z,z18z,z28z,z38z,x11,x2 &1,x31,x12,x22,x32,x13,x23,x33,y11,y21,y31,y12,y22,y32,y13,y23,y33, &z11,z21,z31,z12,z22,z32,z13,z23,z33 common/parm8z/ pi,Omega,Cd,sigma,RT,h0,alpha,e,theta0,g,mode, & rk2,rh2 C############################################################################## C You can solve problems in your region only if you can describe it by # C X = X(P1,P2) # C Y = Y(P1,P2) # C with constant limits on the parameters P1,P2. If your region is # C rectangular, enter ITRANS=0 and the trivial parameterization # C X = P1 # C Y = P2 # C will be used. Otherwise, you need to read the FINE PRINT below. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If P1,P2 represent polar or other non-Cartesian coordinates, you can +# C + reference the Cartesian coordinates X,Y and derivatives of your +# C + unknowns with respect to these coordinates, when you define your +# C + PDE coefficients, boundary conditions, and volume and boundary +# C + integrals, if you enter ITRANS .NE. 0. Enter: +# C + ITRANS = 1, if P1,P2 are polar coordinates, that is, if +# C + P1=R, P2=Theta, where X = R*cos(Theta) +# C + Y = R*sin(Theta) +# C + ITRANS = -1, same as ITRANS=1, but P1=Theta, P2=R +# C + ITRANS = 3, to define your own coordinate transformation. In +# C + this case, you will be prompted to define X,Y and +# C + their first and second derivatives in terms of P1,P2. +# C + Because of symmetry, you will not be prompted for all +# C + of the second derivatives. If you make a mistake in +# C + computing any of these derivatives, PDE2D will usually +# C + be able to issue a warning message. (X1 = dX/dP1, etc) +# C + ITRANS = -3, same as ITRANS=3, but you will only be prompted to +# C + define X,Y; their first and second derivatives will +# C + be approximated using finite differences. +# C + When ITRANS = -3 or 3, the first derivatives of X,Y must all be +# C + continuous. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## ITRANS = 0 C z8z = p38z z38z = 1 return end subroutine pdes8z(yd8z,i8z,j8z,kint8z,p1,p2,p38z,t,uu8z) implicit double precision (a-h,o-z) parameter (neqnmx= 99) C un8z(1,I),un8z(2,I),... hold the (rarely used) values C of UI,UI1,... from the previous iteration or time step common /dtdp5x/un8z(10,neqnmx) common /dtdp18/norm1,norm2,n38z double precision norm1,norm2,n38z,normx,normy,nz8z dimension uu8z(10,neqnmx) common/parm8z/ pi,Omega,Cd,sigma,RT,h0,alpha,e,theta0,g,mode, & rk2,rh2 zr8z = 0.0 U = uu8z(1, 1) U1 = uu8z(2, 1) U2 = uu8z(3, 1) U11= uu8z(5, 1) U22= uu8z(6, 1) U12= uu8z(8, 1) U21= uu8z(8, 1) V = uu8z(1, 2) V1 = uu8z(2, 2) V2 = uu8z(3, 2) V11= uu8z(5, 2) V22= uu8z(6, 2) V12= uu8z(8, 2) V21= uu8z(8, 2) ETA = uu8z(1, 3) ETA1 = uu8z(2, 3) ETA2 = uu8z(3, 3) ETA11= uu8z(5, 3) ETA22= uu8z(6, 3) ETA12= uu8z(8, 3) ETA21= uu8z(8, 3) call dtdpcd(p1,p2,p38z) call dtdpcb(p1,p2,p38z,norm1,norm2,n38z,x,y,z8z,normx,normy,nz8z,3 &) call dtdpcc(p1,p2,p38z, & U1,U2,zr8z,U11,U22,zr8z,U12,zr8z,zr8z, & x,y,z8z,Ux,Uy,uz8z,Uxx,Uyy,uzz8z,Uxy,uxz8z,uyz8z, & Uyx,uzx8z,uzy8z,dvol,darea) Unorm = Ux*normx + Uy*normy call dtdpcc(p1,p2,p38z, & V1,V2,zr8z,V11,V22,zr8z,V12,zr8z,zr8z, & x,y,z8z,Vx,Vy,uz8z,Vxx,Vyy,uzz8z,Vxy,uxz8z,uyz8z, & Vyx,uzx8z,uzy8z,dvol,darea) Vnorm = Vx*normx + Vy*normy call dtdpcc(p1,p2,p38z, & ETA1,ETA2,zr8z,ETA11,ETA22,zr8z,ETA12,zr8z,zr8z, & x,y,z8z,ETAx,ETAy,uz8z,ETAxx,ETAyy,uzz8z,ETAxy,uxz8z,uyz8z, & ETAyx,uzx8z,uzy8z,dvol,darea) ETAnorm = ETAx*normx + ETAy*normy if (i8z.eq.0) then yd8z = 0.0 C############################################################################## C Enter FORTRAN expressions for the functions whose integrals are to be # C calculated and printed. They may be functions of # C # C X,Y,U,Ux,Uy,Uxx,Uyy,Uxy # C V,Vx,Vy,Vxx,Vyy,Vxy # C ETA,ETAx,ETAy,ETAxx,ETAyy,ETAxy and (if applicable) T # C # C The parameters P1,P2 and derivatives with respect to these may also # C be referenced (U1 = dU/dP1, etc): # C U1,U2,U11,U22,U12 # C V1,V2,V11,V22,V12 # C ETA1,ETA2,ETA11,ETA22,ETA12 # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If you only want to integrate a function over part of the region, +# C + define that function to be zero in the rest of the region. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C INTEGRAL DEFINED if (kint8z.eq. 1) yd8z = & Cd*sigma*(u**2+v**2)**1.5*RT**2*cos(y) & + alpha*sigma*h0*(u**2+v**2)*RT**2*cos(y) C############################################################################## C Enter FORTRAN expressions for the functions whose integrals are to be # C calculated and printed. They may be functions of # C # C X,Y,U,Ux,Uy,Uxx,Uyy,Uxy # C V,Vx,Vy,Vxx,Vyy,Vxy # C ETA,ETAx,ETAy,ETAxx,ETAyy,ETAxy and (if applicable) T # C # C The components (NORMx,NORMy) of the unit outward normal vector # C may also be referenced. # C # C The parameters P1,P2 and derivatives with respect to these may also # C be referenced: # C U1,U2,U11,U22,U12 # C V1,V2,V11,V22,V12 # C ETA1,ETA2,ETA11,ETA22,ETA12 # C You can also reference the normal derivatives Unorm,Vnorm,ETAnorm. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If you only want to integrate a function over part of the boundary, +# C + define that function to be zero on the rest of the boundary. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C BND. INTEGRAL1 DEFINED C if (kint8z.eq.-1) yd8z = C & [DEFAULT SELECTED, DEFINITION COMMENTED OUT] if (kint8z.gt.0) yd8z = yd8z*dvol if (kint8z.lt.0) yd8z = yd8z*darea else C############################################################################## C Now enter FORTRAN expressions to define the PDE coefficients, which # C may be functions of # C # C X,Y,T,U,Ux,Uy,Uxx,Uyy,Uxy # C V,Vx,Vy,Vxx,Vyy,Vxy # C ETA,ETAx,ETAy,ETAxx,ETAyy,ETAxy # C # C Recall that the PDEs have the form # C # C C11*d(U)/dT + C12*d(V)/dT + C13*d(ETA)/dT = F1 # C C21*d(U)/dT + C22*d(V)/dT + C23*d(ETA)/dT = F2 # C C31*d(U)/dT + C32*d(V)/dT + C33*d(ETA)/dT = F3 # C # C The parameters P1,P2 and derivatives with respect to these may also # C be referenced (U1 = dU/dP1, etc): # C U1,U2,U11,U22,U12 # C V1,V2,V11,V22,V12 # C ETA1,ETA2,ETA11,ETA22,ETA12 # C############################################################################## c if (mode.eq.1) then c c POT = 0.75*Omega**2*RT**2*e*( (1-3*sin(y)**2)*cos(omega*t) c & + cos(y)**2*(3*cos(2*x)*cos(omega*t)+4*sin(2*x)*sin(omega*t))) POTx = 0.75*Omega**2*RT**2*e*cos(y)**2* & (-6*sin(2*x)*cos(omega*t) + 8*cos(2*x)*sin(omega*t)) POTy = -0.75*Omega**2*RT**2*e*( 6*cos(y)*sin(y)*cos(omega*t) + &2*sin(y)*cos(y)*(3*cos(2*x)*cos(omega*t)+4*sin(2*x)*sin(omega*t))) else c c POT = -1.5*Omega**2*RT**2*theta0*sin(y)*cos(y)* c & (cos(x-Omega*t) + cos(x+Omega*t)) POTx = 1.5*Omega**2*RT**2*theta0*sin(y)*cos(y)* & (sin(x-Omega*t) + sin(x+Omega*t)) POTy = -1.5*Omega**2*RT**2*theta0*(cos(y)**2-sin(y)**2)* & (cos(x-Omega*t) + cos(x+Omega*t)) endif c if (j8z.eq.0) then yd8z = 0.0 C C(1,1) DEFINED if (i8z.eq. -101) yd8z = & 1 C C(1,2) DEFINED if (i8z.eq. -102) yd8z = & 0 C C(1,3) DEFINED if (i8z.eq. -103) yd8z = & 0 C F1 DEFINED if (i8z.eq. 1) yd8z = & 2*Omega*V*sin(y) - Cd/h0*sqrt(u**2+v**2)*u & -alpha*U - g/(RT*cos(y))*ETAx + (1+rk2-rh2)*POTx/(RT*cos(y)) C C(2,1) DEFINED if (i8z.eq. -201) yd8z = & 0 C C(2,2) DEFINED if (i8z.eq. -202) yd8z = & 1 C C(2,3) DEFINED if (i8z.eq. -203) yd8z = & 0 C F2 DEFINED if (i8z.eq. 2) yd8z = & -2*Omega*U*sin(y) - Cd/h0*sqrt(u**2+v**2)*v & -alpha*V - g/RT*ETAy + (1+rk2-rh2)*POTy/RT C C(3,1) DEFINED if (i8z.eq. -301) yd8z = & 0 C C(3,2) DEFINED if (i8z.eq. -302) yd8z = & 0 C C(3,3) DEFINED if (i8z.eq. -303) yd8z = & 1 C F3 DEFINED if (i8z.eq. 3) yd8z = & -h0/(RT*cos(y))*(Ux + cos(y)*Vy - sin(y)*V) else endif endif return end function u8z(i8z,p1,p2,p38z,t0) implicit double precision (a-h,o-z) common/parm8z/ pi,Omega,Cd,sigma,RT,h0,alpha,e,theta0,g,mode, & rk2,rh2 call dtdpcd(p1,p2,p38z) call dtdpcb(p1,p2,p38z,z18z,z28z,z38z,x,y,z8z,d18z,d28z,d38z,1) u8z = 0.0 C############################################################################## C Now the initial values must be defined using FORTRAN expressions. # C They may be functions of X and Y (and the parameters P1,P2), and may # C also reference the initial time T0. # C############################################################################## C U0 DEFINED if (i8z.eq. 1) u8z = & 0 C V0 DEFINED if (i8z.eq. 2) u8z = & 0 C ETA0 DEFINED if (i8z.eq. 3) u8z = & 0 return end subroutine gb8z(gd8z,ifac8z,i8z,j8z,p1,p2,p38z,t,uu8z) implicit double precision (a-h,o-z) parameter (neqnmx= 99) dimension uu8z(10,neqnmx) C un8z(1,I),un8z(2,I),... hold the (rarely used) values C of UI,UI1,... from the previous iteration or time step common /dtdp5x/ un8z(10,neqnmx) common /dtdp18/norm1,norm2,n38z double precision none,norm1,norm2,n38z,normx,normy,nz8z common/parm8z/ pi,Omega,Cd,sigma,RT,h0,alpha,e,theta0,g,mode, & rk2,rh2 none = dtdplx(2) zr8z = 0.0 U = uu8z(1, 1) U1 = uu8z(2, 1) U2 = uu8z(3, 1) V = uu8z(1, 2) V1 = uu8z(2, 2) V2 = uu8z(3, 2) ETA = uu8z(1, 3) ETA1 = uu8z(2, 3) ETA2 = uu8z(3, 3) call dtdpcd(p1,p2,p38z) call dtdpcb(p1,p2,p38z,norm1,norm2,n38z,x,y,z8z,normx,normy,nz8z,3 &) call dtdpcb( & p1,p2,p38z,U1,U2,zr8z,x,y,z8z,Ux,Uy,uz8z,2) Unorm = Ux*normx + Uy*normy call dtdpcb( & p1,p2,p38z,V1,V2,zr8z,x,y,z8z,Vx,Vy,uz8z,2) Vnorm = Vx*normx + Vy*normy call dtdpcb( & p1,p2,p38z,ETA1,ETA2,zr8z,x,y,z8z,ETAx,ETAy,uz8z,2) ETAnorm = ETAx*normx + ETAy*normy if (j8z.eq.0) gd8z = 0.0 C############################################################################## C Enter FORTRAN expressions to define the boundary condition functions, # C which may be functions of # C # C X,Y,U,Ux,Uy, # C V,Vx,Vy, # C ETA,ETAx,ETAy and (if applicable) T # C # C Recall that the boundary conditions have the form # C # C G1 = 0 # C G2 = 0 # C G3 = 0 # C # C Enter NONE to indicate "no" boundary condition. # C # C The parameters P1,P2 and derivatives with respect to these may also # C be referenced (U1 = dU/dP1, etc): # C U1,U2 # C V1,V2 # C ETA1,ETA2 # C The components (NORMx,NORMy) of the unit outward normal vector # C may also be referenced, as well as the normal derivatives Unorm, # C Vnorm,ETAnorm. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If "no" boundary condition is specified, the corresponding PDE is +# C + enforced at points just inside the boundary (exactly on the +# C + boundary, if EPS8Z is set to 0 in the main program). +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## if (ifac8z.eq. 1) then C############################################################################## C # C First define the boundary conditions on the face P1 = P1GRID(1). # C############################################################################## if (j8z.eq.0) then C OmegaIC BOUNDARY CONDITIONS SET (SEE IPERDC) C G1 DEFINED C if (i8z.eq. 1) gd8z = C & [DEFAULT SELECTED, DEFINITION COMMENTED OUT] C G2 DEFINED C if (i8z.eq. 2) gd8z = C & [DEFAULT SELECTED, DEFINITION COMMENTED OUT] C G3 DEFINED C if (i8z.eq. 3) gd8z = C & [DEFAULT SELECTED, DEFINITION COMMENTED OUT] else endif endif if (ifac8z.eq. 2) then C############################################################################## C # C Now define the boundary conditions on the face P1 = P1GRID(NP1GRID). # C############################################################################## if (j8z.eq.0) then C OmegaIC BOUNDARY CONDITIONS SET (SEE IPERDC) C G1 DEFINED C if (i8z.eq. 1) gd8z = C & [DEFAULT SELECTED, DEFINITION COMMENTED OUT] C G2 DEFINED C if (i8z.eq. 2) gd8z = C & [DEFAULT SELECTED, DEFINITION COMMENTED OUT] C G3 DEFINED C if (i8z.eq. 3) gd8z = C & [DEFAULT SELECTED, DEFINITION COMMENTED OUT] else endif endif if (ifac8z.eq. 3) then C############################################################################## C # C Now define the boundary conditions on the face P2 = P2GRID(1). # C############################################################################## if (j8z.eq.0) then C G1 DEFINED if (i8z.eq. 1) gd8z = & none C G2 DEFINED if (i8z.eq. 2) gd8z = & none C G3 DEFINED if (i8z.eq. 3) gd8z = & none else endif endif if (ifac8z.eq. 4) then C############################################################################## C # C Now define the boundary conditions on the face P2 = P2GRID(NP2GRID). # C############################################################################## if (j8z.eq.0) then C G1 DEFINED if (i8z.eq. 1) gd8z = & none C G2 DEFINED if (i8z.eq. 2) gd8z = & none C G3 DEFINED if (i8z.eq. 3) gd8z = & none else endif endif return end subroutine pmod8z(p1,p2,p38z,t,uu8z,uprint,uxprint,uyprint,uzp8z) implicit double precision (a-h,o-z) dimension uu8z(10,*),uprint(*),uxprint(*),uyprint(*),uzp8z(*) common/dtdp14/sint(20),bint(20),slim8z(20),blim8z(20) common/parm8z/ pi,Omega,Cd,sigma,RT,h0,alpha,e,theta0,g,mode, & rk2,rh2 zr8z = 0.0 U = uu8z(1, 1) U1 = uu8z(2, 1) U2 = uu8z(3, 1) U11= uu8z(5, 1) U22= uu8z(6, 1) U12= uu8z(8, 1) U21= uu8z(8, 1) V = uu8z(1, 2) V1 = uu8z(2, 2) V2 = uu8z(3, 2) V11= uu8z(5, 2) V22= uu8z(6, 2) V12= uu8z(8, 2) V21= uu8z(8, 2) ETA = uu8z(1, 3) ETA1 = uu8z(2, 3) ETA2 = uu8z(3, 3) ETA11= uu8z(5, 3) ETA22= uu8z(6, 3) ETA12= uu8z(8, 3) ETA21= uu8z(8, 3) call dtdpcd(p1,p2,p38z) call dtdpcc(p1,p2,p38z, & U1,U2,zr8z,U11,U22,zr8z,U12,zr8z,zr8z, & x,y,z8z,Ux,Uy,uz8z,Uxx,Uyy,uzz8z,Uxy,uxz8z,uyz8z, & Uyx,uzx8z,uzy8z,dvol8z,dare8z) uxprint( 1) = Ux uyprint( 1) = Uy call dtdpcc(p1,p2,p38z, & V1,V2,zr8z,V11,V22,zr8z,V12,zr8z,zr8z, & x,y,z8z,Vx,Vy,uz8z,Vxx,Vyy,uzz8z,Vxy,uxz8z,uyz8z, & Vyx,uzx8z,uzy8z,dvol8z,dare8z) uxprint( 2) = Vx uyprint( 2) = Vy call dtdpcc(p1,p2,p38z, & ETA1,ETA2,zr8z,ETA11,ETA22,zr8z,ETA12,zr8z,zr8z, & x,y,z8z,ETAx,ETAy,uz8z,ETAxx,ETAyy,uzz8z,ETAxy,uxz8z,uyz8z, & ETAyx,uzx8z,uzy8z,dvol8z,dare8z) uxprint( 3) = ETAx uyprint( 3) = ETAy C############################################################################## C If you don't want to read the FINE PRINT, default all of the following # C variables. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + Normally, PDE2D saves the values of U,Ux,Uy,V,Vx,Vy, +# C + ETA,ETAx,ETAy at the output points. If different variables are to +# C + be saved (for later printing or plotting) the following functions +# C + can be used to re-define the output variables: +# C + define UPRINT(1) to replace U +# C + UXPRINT(1) Ux +# C + UYPRINT(1) Uy +# C + UPRINT(2) V +# C + UXPRINT(2) Vx +# C + UYPRINT(2) Vy +# C + UPRINT(3) ETA +# C + UXPRINT(3) ETAx +# C + UYPRINT(3) ETAy +# C + Each function may be a function of +# C + +# C + X,Y,U,Ux,Uy,Uxx,Uyy,Uxy +# C + V,Vx,Vy,Vxx,Vyy,Vxy +# C + ETA,ETAx,ETAy,ETAxx,ETAyy,ETAxy and (if applicable) T +# C + +# C + Each may also be a function of the integral estimates SINT(1),..., +# C + BINT(1),... +# C + +# C + The parameters P1,P2 and derivatives with respect to these may also +# C + be referenced (U1 = dU/dP1, etc): +# C + U1,U2,U11,U22,U12 +# C + V1,V2,V11,V22,V12 +# C + ETA1,ETA2,ETA11,ETA22,ETA12 +# C + +# C + The default for each variable is no change, for example, UPRINT(1) +# C + defaults to U. Enter FORTRAN expressions for each of the +# C + following functions (or default). +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C DEFINE UPRINT(*),UXPRINT(*),UYPRINT(*) HERE: UXPRINT(1) = SINT(1) UYPRINT(1) = & Cd*sigma*(u**2+v**2)**1.5 & + alpha*sigma*h0*(u**2+v**2) return end function axis8z(i8z,p1,p2,p38z,ical8z) implicit double precision (a-h,o-z) call dtdpcd(p1,p2,p38z) call dtdpcb(p1,p2,p38z,z18z,z28z,z38z,x,y,z8z,d18z,d28z,d38z,1) if (i8z.eq.1) axis8z = x if (i8z.eq.2) axis8z = y return end C dummy routines subroutine xy8z(i8z,iarc8z,s,x,y,s0,sf) implicit double precision (a-h,o-z) return end subroutine dis8z(x,y,ktri,triden,shape) implicit double precision (a-h,o-z) return end function fb8z(i8z,iarc8z,ktri,s,x,y,t) implicit double precision (a-h,o-z) fb8z = 0 return end subroutine postpr(tout,nsave,p1out,p2out,np1,np2,uout,neqn, & iprob,nprob) implicit double precision (a-h,o-z) dimension p1out(0:np1,0:np2),p2out(0:np1,0:np2),tout(0:nsave) dimension uout(0:np1,0:np2,4,neqn,0:nsave) common/parm8z/ pi,Omega,Cd,sigma,RT,h0,alpha,e,theta0,g,mode, & rk2,rh2 common /dtdp27/ itask,npes,icomm common /dtdp46/ eps8z,cgtl8z,npmx8z,itype,near8z data lun,lud/0,47/ if (itask.gt.0) return C UOUT(I,J,IDER,IEQ,L) = U_IEQ, if IDER=1 C Ux_IEQ, if IDER=2 C Uy_IEQ, if IDER=3 C (possibly as modified by UPRINT,..) C at the point (P1OUT(I,J) , P2OUT(I,J)) C at time/iteration TOUT(L). C ******* ADD POSTPROCESSING CODE HERE: C IN THE EXAMPLE BELOW, MATLAB PLOTFILES pde2d.m, C pde2d.rdm CREATED (REMOVE COMMENTS TO ACTIVATE) c compute energy dissipation L = 0 do i=1,10 sum = 0 do j=1,20 L = L+1 sum = sum + uout(1,1,2,1,L) enddo sum = sum/20. print *,' average dissipation rate this period (W) = ',sum enddo do i=0,np1 do j=0,np2 uout(i,j,3,2,nsave) = 0 enddo enddo do L=181,200 do i=0,np1 do j=0,np2 uout(i,j,3,2,nsave) = uout(i,j,3,2,nsave) + uout(i,j,3,1,L) enddo enddo enddo do i=0,np1 do j=0,np2 uout(i,j,3,2,nsave) = uout(i,j,3,2,nsave)/20.d0 enddo enddo if (iprob.lt.nprob) return C! if (lun.eq.0) then C! lun = 46 C! open (lun,file='pde2d.m') C! open (lud,file='pde2d.rdm') C! write (lun,*) 'fid = fopen(''pde2d.rdm'');' C! endif C! do 78753 l=0,nsave C! if (tout(l).ne.dtdplx(2)) nsave0 = l C!78753 continue C! write (lud,78754) nsave0 C! write (lud,78754) neqn C! write (lud,78754) np1 C! write (lud,78754) np2 C!78754 format (i8) C! do 78756 i=0,np1 C! do 78755 j=0,np2 C! p1 = p1out(i,j) C! p2 = p2out(i,j) C! p38z = 0.0 C! call dtdpcd(p1,p2,p38z) C! call dtdpcb(p1,p2,p38z,z18z,z28z,z38z,x,y,z8z, C! & d18z,d28z,d38z,1) C! write (lud,78762) p1,p2,x,y C!78755 continue C!78756 continue C! do 78761 l=0,nsave0 C! write (lud,78762) tout(l) C! do 78760 ieq=1,neqn C! do 78759 ider=1,3 C! do 78758 i=0,np1 C! do 78757 j=0,np2 C! write (lud,78762) uout(i,j,ider,ieq,l) C!78757 continue C!78758 continue C!78759 continue C!78760 continue C!78761 continue C!78762 format (e16.8) C ******* WRITE pde2d.m C! call mtdp2dc(itype,lun) return end