C ************************** C * PDE2D 9.4 MAIN PROGRAM * C ************************** C *** 1D PROBLEM SOLVED (COLLOCATION METHOD) *** C############################################################################## C Is double precision mode to be used? Double precision is recommended # C on 32-bit computers. # 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 NXGRID = number of X-grid lines # C############################################################################## PARAMETER (NXGRID = 501) C############################################################################## C How many differential equations (NEQN) are there in your problem? # C############################################################################## PARAMETER (NEQN = 1) C DIMENSIONS OF WORK ARRAYS C SET TO 1 FOR AUTOMATIC ALLOCATION PARAMETER (IRWK8Z= 1) PARAMETER (IIWK8Z= 1) PARAMETER (NXP8Z=1001,KDEG8Z=1) C############################################################################## C The solution is saved on a uniform grid of NX+1 points # C XA + I*(XB-XA)/NX # C I=0,...,NX. Enter a value for NX (suggested value = 50). # 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 NX+1 points, enter -NX. +# 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 or integration point. For most problems this obviously +# C + will produce less accurate output or plots, but for certain (rare) +# C + problems, a solution component may be much less noisy when plotted +# C + only at collocation or integration points. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## PARAMETER (NX = 100) 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 = 1) common/parm8z/ pi,E ,r ,sigma ,Smax ,Tfinal dimension xgrid(nxgrid),xout8z(0:nx),xcross(100),tout8z(0:nsave),u &out(0:nx,2,neqn,0:nsave),xres8z(nxp8z),ures8z(neqn,nxp8z) allocatable iwrk8z(:),rwrk8z(:) C dimension iwrk8z(iiwk8z),rwrk8z(irwk8z) character*40 title logical linear,crankn,noupdt,nodist,fillin,evcmpx,adapt,plot,lsqfi &t,fdiff,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/dtdp42/ nxa8z,kd8z common/dtdp43/ work8z(nxp8z+3) common/dtdp45/ perdc(neqnmx) common/dtdp46/ eps8z,cgtl8z,npmx8z,itype,near8z common/dtdp62/ amin8z(2*neqnmx),amax8z(2*neqnmx) common/dtdp75/ nx18z,xa,xb,uout pi = 4.0*atan(1.d0) nxa8z = nxp8z nx18z = nx+1 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############################################################################## NPROB = 1 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,T,U,Ux)*d(U)/dT = F(X,T,U,Ux,Uxx) # C # C or the steady-state system: # C # C F(X,U,Ux,Uxx) = 0 # C # C or the linear and homogeneous eigenvalue system: # C # C F(X,U,Ux,Uxx) = lambda*RHO(X)*U # C # C with boundary conditions: # C # C G(X,[T],U,Ux) = 0 # C (periodic boundary conditions are also permitted) # C # C at two X values. # C # C For time-dependent problems there are also initial conditions: # C # C U = U0(X) at T=T0 # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# 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 - 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 - REAL(1.0/(1.0+CMPLX(UR,UI)**10)) +# C + F2 = URxx - AIMAG(1.0/(1.0+CMPLX(UR,UI)**10)) +# C + +# C + If your PDEs involve the solution at points other than X, the +# C + function +# C + (D)OLDSOL1(IDER,IEQ,XX,KDEG) +# C + will interpolate (using interpolation of degree KDEG=1,2 or 3) to XX +# C + the function saved in UOUT(*,IDER,IEQ,ISET) on the last time step or +# C + iteration (ISET) for which it has been saved. Thus, for example, if +# C + IDER=1, this will return the latest value of component IEQ of the +# C + solution at XX, assuming this has not been modified using UPRINT... +# C + If your equations involve integrals of the solution, for example, +# C + you can use (D)OLDSOL1 to approximate these using the solution from +# C + the last time step or iteration. +# C + +# C + Note: For a steady-state problem, you must reset NOUT=1 if you want +# C + to save the solution each iteration. +# 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############################################################################## E = & 40.0 r = & 0.1 sigma = & 0.2 Smax = & 100 Tfinal = & 0.5 C############################################################################## C A collocation finite element method is used, with cubic Hermite # C basis functions on the subintervals defined by the grid points: # C XGRID(1),XGRID(2),...,XGRID(NXGRID) # C You will first be prompted for NXGRID, the number of X-grid points, # C then for XGRID(1),...,XGRID(NXGRID). Any points defaulted will be # C uniformly spaced between the points you define; the first and last # C points cannot be defaulted. The interval over which the PDE system # C is to be solved is then: # C XGRID(1) < X < XGRID(NXGRID) # C # C############################################################################## call dtdpwx(xgrid,nxgrid,0) C XGRID DEFINED XGRID(1) = & 0 XGRID(NXGRID/2) = & E XGRID(NXGRID) = & Smax call dtdpwx(xgrid,nxgrid,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 = & Tfinal TF = & 0 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 = .TRUE. 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. TOLER(1) = 0.0001 C############################################################################## C If you don't want to read the FINE PRINT, it is safe (though possibly # C very inefficient) to enter 'no'. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If your time-dependent problem is linear with all PDE and boundary +# C + condition coefficients independent of time except inhomogeneous +# C + terms, then a large savings in execution time may be possible if +# C + this is recognized (the LU decomposition computed on the first step +# C + can be used on subsequent steps). Is this the case for your +# C + problem? (Caution: if you answer 'yes' when you should not, you +# C + will get incorrect results with no warning.) +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## 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############################################################################## NSTEPS = & 50 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 = .TRUE. FDIFF = .FALSE. 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 = 0 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 = .FALSE. C GRIDID = .FALSE. IF FINITE ELEMENT GRID CHANGES BETWEEN DUMP, RESTART GRIDID = .TRUE. C############################################################################## C If you do not have periodic boundary conditions, enter IPERDC=0. # C # C Enter IPERDC=1 for periodic conditions at X = XGRID(1),XGRID(NXGRID) # 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 below in the main program and +# C + set the desired boundary conditions in subroutine GB8Z, "by hand". +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## IPERDC = 0 C############################################################################## C The solution is saved on a uniform grid of NX+1 points, covering the # C interval (XA,XB). Enter values for XA,XB. These variables are usually # C defaulted. # C # C The defaults are XA = XGRID(1), XB = XGRID(NXGRID) # C # C############################################################################## C defaults for xa,xb xa = xgrid(1) xb = xgrid(nxgrid) C DEFINE XA,XB IMMEDIATELY BELOW: call dtdpx1(nx,xa,xb,hx8z,xout8z,npts8z) C *******allocate workspace call dtdp1q(nxgrid,neqn,ii8z,ir8z) if (iiwk8z.gt.1) ii8z = iiwk8z if (irwk8z.gt.1) ir8z = irwk8z allocate (iwrk8z(ii8z),rwrk8z(ir8z)) C *******DRAW GRID POINTS? PLOT = .FALSE. C *******call pde solver call dtdp1x(xgrid, nxgrid, neqn, nint, nbint, xout8z, uout, tout8z &, iperdc, plot, lsqfit, fdiff, npts8z, t0, dt, nsteps, nout, nsave &, crankn, noupdt, itype, linear, rwrk8z, ir8z, iwrk8z, ii8z, restr &t, gridid) deallocate (iwrk8z,rwrk8z) C *******read from restart file to array ures8z C call dtdpr1(1,xres8z,nxp8z,ures8z,neqn) C *******write array ures8z back to restart file C call dtdpr1(2,xres8z,nxp8z,ures8z,neqn) C *******call user-written postprocessor call postpr(tout8z,nsave,xout8z,nx,uout,neqn) C *******LINE PLOTS C############################################################################## C Enter a value for IVAR, to select the variable to be plotted or # C printed: # C IVAR = 1 means V (possibly as modified by UPRINT,..) # C 2 Vx # C############################################################################## IVAR = 1 C X IS VARIABLE ics8z = 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 alow = amin8z(ivar) ahigh = amax8z(ivar) 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 = 0.0 UMAX = 0.0 UMIN = & 0 UMAX = & 60 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 ' call dtdprx(tout8z,nsave,iset1,iset2,isinc) do 78756 is8z=iset1,iset2,isinc call dtdpzp(ics8z,ivar,tout8z,nsave,xout8z,nx,uout,neqn,title,umin &,umax,ix8z,is8z) 78756 continue C *******TABULAR OUTPUT C############################################################################## C Enter a value for IVAR, to select the variable to be plotted or # C printed: # C IVAR = 1 means V (possibly as modified by UPRINT,..) # C 2 Vx # C############################################################################## IVAR = 1 ivara8z = mod(ivar-1,2)+1 ivarb8z = (ivar-1)/2+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 = 0 ISET2 = NSAVE ISINC = 1 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 ' call dtdprx(tout8z,nsave,iset1,iset2,isinc) do 78757 is8z=iset1,iset2,isinc call dtdp1d(xout8z,uout(0,ivara8z,ivarb8z,is8z),npts8z,title,tout8 &z(is8z)) 78757 continue 78755 continue call endgks stop end subroutine pdes8z(yd8z,i8z,j8z,kint8z,x,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,UIx,... from the previous iteration or time step common /dtdp4x/un8z(3,neqnmx) common /dtdp11/normx double precision normx,uu8z(3,neqnmx) common/parm8z/ pi,E ,r ,sigma ,Smax ,Tfinal V = uu8z(1, 1) Vx = uu8z(2, 1) Vxx= uu8z(3, 1) 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,V,Vx,Vxx and (if applicable) T # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If you only want to integrate a function over part of the interval, +# C + define that function to be zero on the rest of the interval. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C INTEGRAL1 DEFINED C if (kint8z.eq.1) yd8z = C & [DEFAULT SELECTED, DEFINITION COMMENTED OUT] C############################################################################## C Enter FORTRAN expressions for the functions whose "integrals" (sum # C over two boundary points) are to be calculated and printed. They may # C be functions of # C # C X,V,Vx,Vxx and (if applicable) T # C # C The unit outward normal, NORMx (=1 at right endpoint, -1 at left), # C may also be referenced. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If you only want to "integrate" a function over one boundary point, +# C + define that function to be zero at the other point. +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C BND. INTEGRAL1 DEFINED C if (kint8z.eq.-1) yd8z = C & [DEFAULT SELECTED, DEFINITION COMMENTED OUT] else C############################################################################## C Now enter FORTRAN expressions to define the PDE coefficients, which # C may be functions of # C # C X,T,V,Vx,Vxx # C # C Recall that the PDE has the form # C # C C*d(V)/dT = F # C # C############################################################################## if (j8z.eq.0) then yd8z = 0.0 C C DEFINED if (i8z.eq. -101) yd8z = & -1 C F DEFINED if (i8z.eq. 1) yd8z = & 0.5*sigma**2*x**2*Vxx + r*x*Vx - r*V else endif endif return end function u8z(i8z,x,t0) implicit double precision (a-h,o-z) common/parm8z/ pi,E ,r ,sigma ,Smax ,Tfinal u8z = 0.0 C############################################################################## C Now the initial values must be defined using FORTRAN expressions. # C They may be functions of X, and may also reference the initial time T0. # C############################################################################## C V0 DEFINED if (i8z.eq. 1) u8z = & max(0.d0,x-E) return end subroutine gb8z(gd8z,ifac8z,i8z,j8z,x,t,uu8z) implicit double precision (a-h,o-z) parameter (neqnmx= 99) dimension uu8z(3,neqnmx) C un8z(1,I),un8z(2,I),... hold the (rarely used) values C of UI,UIx,... from the previous iteration or time step common /dtdp4x/ un8z(3,neqnmx) double precision none common/parm8z/ pi,E ,r ,sigma ,Smax ,Tfinal none = dtdplx(2) V = uu8z(1, 1) Vx = uu8z(2, 1) 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,V,Vx and (if applicable) T # C # C Recall that the boundary conditions have the form # C # C G = 0 # C # C Enter NONE to indicate "no" boundary condition. # C # C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++# C + If "no" boundary condition is specified, the PDE is enforced at a +# C + point just inside the boundary (exactly on the boundary, if EPS8Z +# C + 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 at the point X = XGRID(1). # C############################################################################## if (j8z.eq.0) then C G DEFINED if (i8z.eq. 1) gd8z = & V else endif endif if (ifac8z.eq. 2) then C############################################################################## C # C Now define the boundary conditions at the point X = XGRID(NXGRID). # C############################################################################## if (j8z.eq.0) then C G DEFINED if (i8z.eq. 1) gd8z = & Vx-1 else endif endif return end subroutine pmod8z(x,t,uu8z,uprint,uxprnt) implicit double precision (a-h,o-z) dimension uu8z(3,*),uprint(*),uxprnt(*) common/dtdp14/sint(20),bint(20),slim8z(20),blim8z(20) common/parm8z/ pi,E ,r ,sigma ,Smax ,Tfinal V = uu8z(1, 1) Vx = uu8z(2, 1) Vxx= uu8z(3, 1) 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 V,Vx at the output points. +# C + If different variables are to be saved (for later printing or +# C + plotting) the following functions can be used to re-define the +# C + output variables: +# C + define UPRINT(1) to replace V +# C + UXPRNT(1) Vx +# C + Each function may be a function of +# C + +# C + X,V,Vx,Vxx and (if applicable) T +# C + +# C + Each may also be a function of the integral estimates SINT(1),..., +# C + BINT(1),... +# C + +# C + The default for each variable is no change, for example, UPRINT(1) +# C + defaults to V. Enter FORTRAN expressions for each of the +# C + following functions (or default). +# C ++++++++++++++++++++++++++ END OF "FINE PRINT" +++++++++++++++++++++++++# C############################################################################## C DEFINE UPRINT(*),UXPRNT(*) HERE: 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 function axis8z(i8z,x,y,z,ical8z) implicit double precision (a-h,o-z) axis8z = 0 return end subroutine tran8z(itrans,x,y,z) implicit double precision (a-h,o-z) return end subroutine postpr(tout,nsave,xout,nx,uout,neqn) implicit double precision (a-h,o-z) dimension xout(0:nx),tout(0:nsave) dimension uout(0:nx,2,neqn,0:nsave) common/parm8z/ pi,E ,r ,sigma ,Smax ,Tfinal 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,IDER,IEQ,L) = U_IEQ, if IDER=1 C Ux_IEQ, if IDER=2 C (possibly as modified by UPRINT,..) C at the point XOUT(I) 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) if (lun.eq.0) then lun = 46 open (lun,file='pde2d.m') open (lud,file='pde2d.rdm') write (lun,*) 'fid = fopen(''pde2d.rdm'');' endif do 78753 l=0,nsave if (tout(l).ne.dtdplx(2)) nsave0 = l 78753 continue write (lud,78754) nsave0 write (lud,78754) neqn write (lud,78754) nx 78754 format (i8) do 78755 i=0,nx write (lud,78760) xout(i) 78755 continue do 78759 l=0,nsave0 write (lud,78760) tout(l) do 78758 ieq=1,neqn do 78757 ider=1,2 do 78756 i=0,nx write (lud,78760) uout(i,ider,ieq,l) 78756 continue 78757 continue 78758 continue 78759 continue 78760 format (e16.8) write (lun,*) '% Read solution from pde2d.rdm' write (lun,*) 'NSAVE = fscanf(fid,''%g'',1);' write (lun,*) 'NEQN = fscanf(fid,''%g'',1);' write (lun,*) 'NX = fscanf(fid,''%g'',1);' if (itype.eq.2) then write (lun,*) 'L0 = 0;' else write (lun,*) 'L0 = 1;' endif write (lun,*) 'T = zeros(NSAVE+1,1);' write (lun,*) 'X = zeros(NX+1,1);' write (lun,*) 'U = zeros(NX+1,NSAVE+1,2,NEQN);' write (lun,*) 'for i=0:NX' write (lun,*) ' X(i+1) = fscanf(fid,''%g'',1);' write (lun,*) 'end' write (lun,*) 'for l=0:NSAVE' write (lun,*) 'T(l+1) = fscanf(fid,''%g'',1);' write (lun,*) 'for ieq=1:NEQN' write (lun,*) 'for ider=1:2' write (lun,*) 'for i=0:NX' write (lun,*) & ' U(i+1,l+1,ider,ieq) = fscanf(fid,''%g'',1);' write (lun,*) 'end' write (lun,*) 'end' write (lun,*) 'end' write (lun,*) 'end' write (lun,*) 'xmin = min(X(:));' write (lun,*) 'xmax = max(X(:));' write (lun,*) '% Plots of each variable' write (lun,*) 'for IEQ=1:NEQN' write (lun,*) '% Plot U_IEQ, if IDER=1' write (lun,*) '% Ux_IEQ, if IDER=2' write (lun,*) 'IDER = 1;' write (lun,*) 'umin = min(min(U(:,L0+1:NSAVE+1,IDER,IEQ)));' write (lun,*) 'umax = max(max(U(:,L0+1:NSAVE+1,IDER,IEQ)));' write (lun,*) 'for L=L0:NSAVE' write (lun,*) ' figure' write (lun,*) ' plot(X,U(:,L+1,IDER,IEQ))' write (lun,*) ' axis([xmin xmax umin umax])' write (lun,*) ' xlabel(''X'')' write (lun,*) ' ylabel([''U'',num2str(IEQ)])' write (lun,*) ' title(['' T = '',num2str(T(L+1))])' write (lun,*) 'end' write (lun,*) 'end' return end