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* -----------------------------------------------------------------
* $Revision: 1.2 $
* $Date: 2010/12/01 22:51:32 $
* -----------------------------------------------------------------
* Programmer(s): Scott D. Cohen, Alan C. Hindmarsh and
* Radu Serban @LLNL
* -----------------------------------------------------------------
* Example problem:
*
* An ODE system is generated from the following 2-species diurnal
* kinetics advection-diffusion PDE system in 2 space dimensions:
*
* dc(i)/dt = Kh*(d/dx)^2 c(i) + V*dc(i)/dx + (d/dy)(Kv(y)*dc(i)/dy)
* + Ri(c1,c2,t) for i = 1,2, where
* R1(c1,c2,t) = -q1*c1*c3 - q2*c1*c2 + 2*q3(t)*c3 + q4(t)*c2 ,
* R2(c1,c2,t) = q1*c1*c3 - q2*c1*c2 - q4(t)*c2 ,
* Kv(y) = Kv0*exp(y/5) ,
* Kh, V, Kv0, q1, q2, and c3 are constants, and q3(t) and q4(t)
* vary diurnally. The problem is posed on the square
* 0 <= x <= 20, 30 <= y <= 50 (all in km),
* with homogeneous Neumann boundary conditions, and for time t in
* 0 <= t <= 86400 sec (1 day).
* The PDE system is treated by central differences on a uniform
* 10 x 10 mesh, with simple polynomial initial profiles.
* The problem is solved with CVODE, with the BDF/GMRES
* method (i.e. using the CVSPGMR linear solver) and a banded
* preconditioner, generated by difference quotients, using the
* module CVBANDPRE. The problem is solved with left and right
* preconditioning.
* -----------------------------------------------------------------
*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <cvode/cvode.h> /* main integrator header file */
#include <cvode/cvode_spgmr.h> /* prototypes & constants for CVSPGMR solver */
#include <cvode/cvode_bandpre.h> /* prototypes & constants for CVBANDPRE module */
#include <nvector/nvector_serial.h> /* serial N_Vector types, fct. and macros */
#include <sundials/sundials_types.h> /* definition of realtype */
#include <sundials/sundials_math.h> /* contains the macros ABS, SQR, and EXP */
/* Problem Constants */
#define ZERO RCONST(0.0)
#define ONE RCONST(1.0)
#define TWO RCONST(2.0)
#define NUM_SPECIES 2 /* number of species */
#define KH RCONST(4.0e-6) /* horizontal diffusivity Kh */
#define VEL RCONST(0.001) /* advection velocity V */
#define KV0 RCONST(1.0e-8) /* coefficient in Kv(y) */
#define Q1 RCONST(1.63e-16) /* coefficients q1, q2, c3 */
#define Q2 RCONST(4.66e-16)
#define C3 RCONST(3.7e16)
#define A3 RCONST(22.62) /* coefficient in expression for q3(t) */
#define A4 RCONST(7.601) /* coefficient in expression for q4(t) */
#define C1_SCALE RCONST(1.0e6) /* coefficients in initial profiles */
#define C2_SCALE RCONST(1.0e12)
#define T0 ZERO /* initial time */
#define NOUT 12 /* number of output times */
#define TWOHR RCONST(7200.0) /* number of seconds in two hours */
#define HALFDAY RCONST(4.32e4) /* number of seconds in a half day */
#define PI RCONST(3.1415926535898) /* pi */
#define XMIN ZERO /* grid boundaries in x */
#define XMAX RCONST(20.0)
#define YMIN RCONST(30.0) /* grid boundaries in y */
#define YMAX RCONST(50.0)
#define XMID RCONST(10.0) /* grid midpoints in x,y */
#define YMID RCONST(40.0)
#define MX 10 /* MX = number of x mesh points */
#define MY 10 /* MY = number of y mesh points */
#define NSMX 20 /* NSMX = NUM_SPECIES*MX */
#define MM (MX*MY) /* MM = MX*MY */
/* CVodeInit Constants */
#define RTOL RCONST(1.0e-5) /* scalar relative tolerance */
#define FLOOR RCONST(100.0) /* value of C1 or C2 at which tolerances */
/* change from relative to absolute */
#define ATOL (RTOL*FLOOR) /* scalar absolute tolerance */
#define NEQ (NUM_SPECIES*MM) /* NEQ = number of equations */
/* User-defined vector and matrix accessor macros: IJKth, IJth */
/* IJKth is defined in order to isolate the translation from the
mathematical 3-dimensional structure of the dependent variable vector
to the underlying 1-dimensional storage. IJth is defined in order to
write code which indexes into small dense matrices with a (row,column)
pair, where 1 <= row, column <= NUM_SPECIES.
IJKth(vdata,i,j,k) references the element in the vdata array for
species i at mesh point (j,k), where 1 <= i <= NUM_SPECIES,
0 <= j <= MX-1, 0 <= k <= MY-1. The vdata array is obtained via
the macro call vdata = NV_DATA_S(v), where v is an N_Vector.
For each mesh point (j,k), the elements for species i and i+1 are
contiguous within vdata.
IJth(a,i,j) references the (i,j)th entry of the small matrix realtype **a,
where 1 <= i,j <= NUM_SPECIES. The small matrix routines in dense.h
work with matrices stored by column in a 2-dimensional array. In C,
arrays are indexed starting at 0, not 1. */
#define IJKth(vdata,i,j,k) (vdata[i-1 + (j)*NUM_SPECIES + (k)*NSMX])
#define IJth(a,i,j) (a[j-1][i-1])
/* Type : UserData
contains preconditioner blocks, pivot arrays, and problem constants */
typedef struct {
realtype q4, om, dx, dy, hdco, haco, vdco;
} *UserData;
/* Private Helper Functions */
static void InitUserData(UserData data);
static void SetInitialProfiles(N_Vector u, realtype dx, realtype dy);
static void PrintIntro(long int mu, long int ml);
static void PrintOutput(void *cvode_mem, N_Vector u, realtype t);
static void PrintFinalStats(void *cvode_mem);
/* Private function to check function return values */
static int check_flag(void *flagvalue, char *funcname, int opt);
/* Function Called by the Solver */
static int f(realtype t, N_Vector u, N_Vector udot, void *user_data);
/*
*-------------------------------
* Main Program
*-------------------------------
*/
int main()
{
realtype abstol, reltol, t, tout;
N_Vector u;
UserData data;
void *cvode_mem;
int flag, iout, jpre;
long int ml, mu;
u = NULL;
data = NULL;
cvode_mem = NULL;
/* Allocate and initialize u, and set problem data and tolerances */
u = N_VNew_Serial(NEQ);
if(check_flag((void *)u, "N_VNew_Serial", 0)) return(1);
data = (UserData) malloc(sizeof *data);
if(check_flag((void *)data, "malloc", 2)) return(1);
InitUserData(data);
SetInitialProfiles(u, data->dx, data->dy);
abstol = ATOL;
reltol = RTOL;
/* Call CVodeCreate to create the solver memory and specify the
* Backward Differentiation Formula and the use of a Newton iteration */
cvode_mem = CVodeCreate(CV_BDF, CV_NEWTON);
if(check_flag((void *)cvode_mem, "CVodeCreate", 0)) return(1);
/* Set the pointer to user-defined data */
flag = CVodeSetUserData(cvode_mem, data);
if(check_flag(&flag, "CVodeSetUserData", 1)) return(1);
/* Call CVodeInit to initialize the integrator memory and specify the
* user's right hand side function in u'=f(t,u), the inital time T0, and
* the initial dependent variable vector u. */
flag = CVodeInit(cvode_mem, f, T0, u);
if(check_flag(&flag, "CVodeInit", 1)) return(1);
/* Call CVodeSStolerances to specify the scalar relative tolerance
* and scalar absolute tolerances */
flag = CVodeSStolerances(cvode_mem, reltol, abstol);
if (check_flag(&flag, "CVodeSStolerances", 1)) return(1);
/* Call CVSpgmr to specify the linear solver CVSPGMR
with left preconditioning and the maximum Krylov dimension maxl */
flag = CVSpgmr(cvode_mem, PREC_LEFT, 0);
if(check_flag(&flag, "CVSpgmr", 1)) return(1);
/* Call CVBandPreInit to initialize band preconditioner */
ml = mu = 2;
flag = CVBandPrecInit(cvode_mem, NEQ, mu, ml);
if(check_flag(&flag, "CVBandPrecInit", 0)) return(1);
PrintIntro(mu, ml);
/* Loop over jpre (= PREC_LEFT, PREC_RIGHT), and solve the problem */
for (jpre = PREC_LEFT; jpre <= PREC_RIGHT; jpre++) {
/* On second run, re-initialize u, the solver, and CVSPGMR */
if (jpre == PREC_RIGHT) {
SetInitialProfiles(u, data->dx, data->dy);
flag = CVodeReInit(cvode_mem, T0, u);
if(check_flag(&flag, "CVodeReInit", 1)) return(1);
flag = CVSpilsSetPrecType(cvode_mem, PREC_RIGHT);
check_flag(&flag, "CVSpilsSetPrecType", 1);
printf("\n\n-------------------------------------------------------");
printf("------------\n");
}
printf("\n\nPreconditioner type is: jpre = %s\n\n",
(jpre == PREC_LEFT) ? "PREC_LEFT" : "PREC_RIGHT");
/* In loop over output points, call CVode, print results, test for error */
for (iout = 1, tout = TWOHR; iout <= NOUT; iout++, tout += TWOHR) {
flag = CVode(cvode_mem, tout, u, &t, CV_NORMAL);
check_flag(&flag, "CVode", 1);
PrintOutput(cvode_mem, u, t);
if (flag != CV_SUCCESS) {
break;
}
}
/* Print final statistics */
PrintFinalStats(cvode_mem);
} /* End of jpre loop */
/* Free memory */
N_VDestroy_Serial(u);
free(data);
CVodeFree(&cvode_mem);
return(0);
}
/*
*-------------------------------
* Private helper functions
*-------------------------------
*/
/* Load problem constants in data */
static void InitUserData(UserData data)
{
data->om = PI/HALFDAY;
data->dx = (XMAX-XMIN)/(MX-1);
data->dy = (YMAX-YMIN)/(MY-1);
data->hdco = KH/SQR(data->dx);
data->haco = VEL/(TWO*data->dx);
data->vdco = (ONE/SQR(data->dy))*KV0;
}
/* Set initial conditions in u */
static void SetInitialProfiles(N_Vector u, realtype dx, realtype dy)
{
int jx, jy;
realtype x, y, cx, cy;
realtype *udata;
/* Set pointer to data array in vector u. */
udata = NV_DATA_S(u);
/* Load initial profiles of c1 and c2 into u vector */
for (jy = 0; jy < MY; jy++) {
y = YMIN + jy*dy;
cy = SQR(RCONST(0.1)*(y - YMID));
cy = ONE - cy + RCONST(0.5)*SQR(cy);
for (jx = 0; jx < MX; jx++) {
x = XMIN + jx*dx;
cx = SQR(RCONST(0.1)*(x - XMID));
cx = ONE - cx + RCONST(0.5)*SQR(cx);
IJKth(udata,1,jx,jy) = C1_SCALE*cx*cy;
IJKth(udata,2,jx,jy) = C2_SCALE*cx*cy;
}
}
}
static void PrintIntro(long int mu, long int ml)
{
printf("2-species diurnal advection-diffusion problem, %d by %d mesh\n",
MX, MY);
printf("SPGMR solver; band preconditioner; mu = %d, ml = %d\n\n",
mu, ml);
return;
}
/* Print current t, step count, order, stepsize, and sampled c1,c2 values */
static void PrintOutput(void *cvode_mem, N_Vector u,realtype t)
{
long int nst;
int qu, flag;
realtype hu, *udata;
int mxh = MX/2 - 1, myh = MY/2 - 1, mx1 = MX - 1, my1 = MY - 1;
udata = NV_DATA_S(u);
flag = CVodeGetNumSteps(cvode_mem, &nst);
check_flag(&flag, "CVodeGetNumSteps", 1);
flag = CVodeGetLastOrder(cvode_mem, &qu);
check_flag(&flag, "CVodeGetLastOrder", 1);
flag = CVodeGetLastStep(cvode_mem, &hu);
check_flag(&flag, "CVodeGetLastStep", 1);
#if defined(SUNDIALS_EXTENDED_PRECISION)
printf("t = %.2Le no. steps = %ld order = %d stepsize = %.2Le\n",
t, nst, qu, hu);
printf("c1 (bot.left/middle/top rt.) = %12.3Le %12.3Le %12.3Le\n",
IJKth(udata,1,0,0), IJKth(udata,1,mxh,myh), IJKth(udata,1,mx1,my1));
printf("c2 (bot.left/middle/top rt.) = %12.3Le %12.3Le %12.3Le\n\n",
IJKth(udata,2,0,0), IJKth(udata,2,mxh,myh), IJKth(udata,2,mx1,my1));
#elif defined(SUNDIALS_DOUBLE_PRECISION)
printf("t = %.2le no. steps = %ld order = %d stepsize = %.2le\n",
t, nst, qu, hu);
printf("c1 (bot.left/middle/top rt.) = %12.3le %12.3le %12.3le\n",
IJKth(udata,1,0,0), IJKth(udata,1,mxh,myh), IJKth(udata,1,mx1,my1));
printf("c2 (bot.left/middle/top rt.) = %12.3le %12.3le %12.3le\n\n",
IJKth(udata,2,0,0), IJKth(udata,2,mxh,myh), IJKth(udata,2,mx1,my1));
#else
printf("t = %.2e no. steps = %ld order = %d stepsize = %.2e\n",
t, nst, qu, hu);
printf("c1 (bot.left/middle/top rt.) = %12.3e %12.3e %12.3e\n",
IJKth(udata,1,0,0), IJKth(udata,1,mxh,myh), IJKth(udata,1,mx1,my1));
printf("c2 (bot.left/middle/top rt.) = %12.3e %12.3e %12.3e\n\n",
IJKth(udata,2,0,0), IJKth(udata,2,mxh,myh), IJKth(udata,2,mx1,my1));
#endif
}
/* Get and print final statistics */
static void PrintFinalStats(void *cvode_mem)
{
long int lenrw, leniw ;
long int lenrwLS, leniwLS;
long int lenrwBP, leniwBP;
long int nst, nfe, nsetups, nni, ncfn, netf;
long int nli, npe, nps, ncfl, nfeLS;
long int nfeBP;
int flag;
flag = CVodeGetWorkSpace(cvode_mem, &lenrw, &leniw);
check_flag(&flag, "CVodeGetWorkSpace", 1);
flag = CVodeGetNumSteps(cvode_mem, &nst);
check_flag(&flag, "CVodeGetNumSteps", 1);
flag = CVodeGetNumRhsEvals(cvode_mem, &nfe);
check_flag(&flag, "CVodeGetNumRhsEvals", 1);
flag = CVodeGetNumLinSolvSetups(cvode_mem, &nsetups);
check_flag(&flag, "CVodeGetNumLinSolvSetups", 1);
flag = CVodeGetNumErrTestFails(cvode_mem, &netf);
check_flag(&flag, "CVodeGetNumErrTestFails", 1);
flag = CVodeGetNumNonlinSolvIters(cvode_mem, &nni);
check_flag(&flag, "CVodeGetNumNonlinSolvIters", 1);
flag = CVodeGetNumNonlinSolvConvFails(cvode_mem, &ncfn);
check_flag(&flag, "CVodeGetNumNonlinSolvConvFails", 1);
flag = CVSpilsGetWorkSpace(cvode_mem, &lenrwLS, &leniwLS);
check_flag(&flag, "CVSpilsGetWorkSpace", 1);
flag = CVSpilsGetNumLinIters(cvode_mem, &nli);
check_flag(&flag, "CVSpilsGetNumLinIters", 1);
flag = CVSpilsGetNumPrecEvals(cvode_mem, &npe);
check_flag(&flag, "CVSpilsGetNumPrecEvals", 1);
flag = CVSpilsGetNumPrecSolves(cvode_mem, &nps);
check_flag(&flag, "CVSpilsGetNumPrecSolves", 1);
flag = CVSpilsGetNumConvFails(cvode_mem, &ncfl);
check_flag(&flag, "CVSpilsGetNumConvFails", 1);
flag = CVSpilsGetNumRhsEvals(cvode_mem, &nfeLS);
check_flag(&flag, "CVSpilsGetNumRhsEvals", 1);
flag = CVBandPrecGetWorkSpace(cvode_mem, &lenrwBP, &leniwBP);
check_flag(&flag, "CVBandPrecGetWorkSpace", 1);
flag = CVBandPrecGetNumRhsEvals(cvode_mem, &nfeBP);
check_flag(&flag, "CVBandPrecGetNumRhsEvals", 1);
printf("\nFinal Statistics.. \n\n");
printf("lenrw = %5ld leniw = %5ld\n", lenrw, leniw);
printf("lenrwls = %5ld leniwls = %5ld\n", lenrwLS, leniwLS);
printf("lenrwbp = %5ld leniwbp = %5ld\n", lenrwBP, leniwBP);
printf("nst = %5ld\n" , nst);
printf("nfe = %5ld nfetot = %5ld\n" , nfe, nfe+nfeLS+nfeBP);
printf("nfeLS = %5ld nfeBP = %5ld\n" , nfeLS, nfeBP);
printf("nni = %5ld nli = %5ld\n" , nni, nli);
printf("nsetups = %5ld netf = %5ld\n" , nsetups, netf);
printf("npe = %5ld nps = %5ld\n" , npe, nps);
printf("ncfn = %5ld ncfl = %5ld\n\n", ncfn, ncfl);
}
/* Check function return value...
opt == 0 means SUNDIALS function allocates memory so check if
returned NULL pointer
opt == 1 means SUNDIALS function returns a flag so check if
flag >= 0
opt == 2 means function allocates memory so check if returned
NULL pointer */
static int check_flag(void *flagvalue, char *funcname, int opt)
{
int *errflag;
/* Check if SUNDIALS function returned NULL pointer - no memory allocated */
if (opt == 0 && flagvalue == NULL) {
fprintf(stderr, "\nSUNDIALS_ERROR: %s() failed - returned NULL pointer\n\n",
funcname);
return(1); }
/* Check if flag < 0 */
else if (opt == 1) {
errflag = (int *) flagvalue;
if (*errflag < 0) {
fprintf(stderr, "\nSUNDIALS_ERROR: %s() failed with flag = %d\n\n",
funcname, *errflag);
return(1); }}
/* Check if function returned NULL pointer - no memory allocated */
else if (opt == 2 && flagvalue == NULL) {
fprintf(stderr, "\nMEMORY_ERROR: %s() failed - returned NULL pointer\n\n",
funcname);
return(1); }
return(0);
}
/*
*-------------------------------
* Function called by the solver
*-------------------------------
*/
/* f routine. Compute f(t,u). */
static int f(realtype t, N_Vector u, N_Vector udot,void *user_data)
{
realtype q3, c1, c2, c1dn, c2dn, c1up, c2up, c1lt, c2lt;
realtype c1rt, c2rt, cydn, cyup, hord1, hord2, horad1, horad2;
realtype qq1, qq2, qq3, qq4, rkin1, rkin2, s, vertd1, vertd2, ydn, yup;
realtype q4coef, dely, verdco, hordco, horaco;
realtype *udata, *dudata;
int idn, iup, ileft, iright, jx, jy;
UserData data;
data = (UserData) user_data;
udata = NV_DATA_S(u);
dudata = NV_DATA_S(udot);
/* Set diurnal rate coefficients. */
s = sin(data->om*t);
if (s > ZERO) {
q3 = EXP(-A3/s);
data->q4 = EXP(-A4/s);
} else {
q3 = ZERO;
data->q4 = ZERO;
}
/* Make local copies of problem variables, for efficiency. */
q4coef = data->q4;
dely = data->dy;
verdco = data->vdco;
hordco = data->hdco;
horaco = data->haco;
/* Loop over all grid points. */
for (jy = 0; jy < MY; jy++) {
/* Set vertical diffusion coefficients at jy +- 1/2 */
ydn = YMIN + (jy - RCONST(0.5))*dely;
yup = ydn + dely;
cydn = verdco*EXP(RCONST(0.2)*ydn);
cyup = verdco*EXP(RCONST(0.2)*yup);
idn = (jy == 0) ? 1 : -1;
iup = (jy == MY-1) ? -1 : 1;
for (jx = 0; jx < MX; jx++) {
/* Extract c1 and c2, and set kinetic rate terms. */
c1 = IJKth(udata,1,jx,jy);
c2 = IJKth(udata,2,jx,jy);
qq1 = Q1*c1*C3;
qq2 = Q2*c1*c2;
qq3 = q3*C3;
qq4 = q4coef*c2;
rkin1 = -qq1 - qq2 + TWO*qq3 + qq4;
rkin2 = qq1 - qq2 - qq4;
/* Set vertical diffusion terms. */
c1dn = IJKth(udata,1,jx,jy+idn);
c2dn = IJKth(udata,2,jx,jy+idn);
c1up = IJKth(udata,1,jx,jy+iup);
c2up = IJKth(udata,2,jx,jy+iup);
vertd1 = cyup*(c1up - c1) - cydn*(c1 - c1dn);
vertd2 = cyup*(c2up - c2) - cydn*(c2 - c2dn);
/* Set horizontal diffusion and advection terms. */
ileft = (jx == 0) ? 1 : -1;
iright =(jx == MX-1) ? -1 : 1;
c1lt = IJKth(udata,1,jx+ileft,jy);
c2lt = IJKth(udata,2,jx+ileft,jy);
c1rt = IJKth(udata,1,jx+iright,jy);
c2rt = IJKth(udata,2,jx+iright,jy);
hord1 = hordco*(c1rt - TWO*c1 + c1lt);
hord2 = hordco*(c2rt - TWO*c2 + c2lt);
horad1 = horaco*(c1rt - c1lt);
horad2 = horaco*(c2rt - c2lt);
/* Load all terms into udot. */
IJKth(dudata, 1, jx, jy) = vertd1 + hord1 + horad1 + rkin1;
IJKth(dudata, 2, jx, jy) = vertd2 + hord2 + horad2 + rkin2;
}
}
return(0);
}
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