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* -----------------------------------------------------------------
* $Revision: 1.2 $
* $Date: 2009/09/30 23:25:59 $
* -----------------------------------------------------------------
* Programmer(s): Allan Taylor, Alan Hindmarsh and
* Radu Serban @ LLNL
* -----------------------------------------------------------------
* Example program for IDA: Food web problem.
*
* This example program (serial version) uses the IDABAND linear
* solver, and IDACalcIC for initial condition calculation.
*
* The mathematical problem solved in this example is a DAE system
* that arises from a system of partial differential equations after
* spatial discretization. The PDE system is a food web population
* model, with predator-prey interaction and diffusion on the unit
* square in two dimensions. The dependent variable vector is:
*
* 1 2 ns
* c = (c , c , ..., c ) , ns = 2 * np
*
* and the PDE's are as follows:
*
* i i i
* dc /dt = d(i)*(c + c ) + R (x,y,c) (i = 1,...,np)
* xx yy i
*
* i i
* 0 = d(i)*(c + c ) + R (x,y,c) (i = np+1,...,ns)
* xx yy i
*
* where the reaction terms R are:
*
* i ns j
* R (x,y,c) = c * (b(i) + sum a(i,j)*c )
* i j=1
*
* The number of species is ns = 2 * np, with the first np being
* prey and the last np being predators. The coefficients a(i,j),
* b(i), d(i) are:
*
* a(i,i) = -AA (all i)
* a(i,j) = -GG (i <= np , j > np)
* a(i,j) = EE (i > np, j <= np)
* all other a(i,j) = 0
* b(i) = BB*(1+ alpha * x*y + beta*sin(4 pi x)*sin(4 pi y)) (i <= np)
* b(i) =-BB*(1+ alpha * x*y + beta*sin(4 pi x)*sin(4 pi y)) (i > np)
* d(i) = DPREY (i <= np)
* d(i) = DPRED (i > np)
*
* The various scalar parameters required are set using '#define'
* statements or directly in routine InitUserData. In this program,
* np = 1, ns = 2. The boundary conditions are homogeneous Neumann:
* normal derivative = 0.
*
* A polynomial in x and y is used to set the initial values of the
* first np variables (the prey variables) at each x,y location,
* while initial values for the remaining (predator) variables are
* set to a flat value, which is corrected by IDACalcIC.
*
* The PDEs are discretized by central differencing on a MX by MY
* mesh.
*
* The DAE system is solved by IDA using the IDABAND linear solver.
* Output is printed at t = 0, .001, .01, .1, .4, .7, 1.
* -----------------------------------------------------------------
* References:
* [1] Peter N. Brown and Alan C. Hindmarsh,
* Reduced Storage Matrix Methods in Stiff ODE systems, Journal
* of Applied Mathematics and Computation, Vol. 31 (May 1989),
* pp. 40-91.
*
* [2] Peter N. Brown, Alan C. Hindmarsh, and Linda R. Petzold,
* Using Krylov Methods in the Solution of Large-Scale
* Differential-Algebraic Systems, SIAM J. Sci. Comput., 15
* (1994), pp. 1467-1488.
*
* [3] Peter N. Brown, Alan C. Hindmarsh, and Linda R. Petzold,
* Consistent Initial Condition Calculation for Differential-
* Algebraic Systems, SIAM J. Sci. Comput., 19 (1998),
* pp. 1495-1512.
* -----------------------------------------------------------------
*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <ida/ida.h>
#include <ida/ida_band.h>
#include <nvector/nvector_serial.h>
#include <sundials/sundials_dense.h>
#include <sundials/sundials_types.h>
/* Problem Constants. */
#define NPREY 1 /* No. of prey (= no. of predators). */
#define NUM_SPECIES 2*NPREY
#define PI RCONST(3.1415926535898)
#define FOURPI (RCONST(4.0)*PI)
#define MX 20 /* MX = number of x mesh points */
#define MY 20 /* MY = number of y mesh points */
#define NSMX (NUM_SPECIES * MX)
#define NEQ (NUM_SPECIES*MX*MY)
#define AA RCONST(1.0) /* Coefficient in above eqns. for a */
#define EE RCONST(10000.) /* Coefficient in above eqns. for a */
#define GG RCONST(0.5e-6) /* Coefficient in above eqns. for a */
#define BB RCONST(1.0) /* Coefficient in above eqns. for b */
#define DPREY RCONST(1.0) /* Coefficient in above eqns. for d */
#define DPRED RCONST(0.05) /* Coefficient in above eqns. for d */
#define ALPHA RCONST(50.) /* Coefficient alpha in above eqns. */
#define BETA RCONST(1000.) /* Coefficient beta in above eqns. */
#define AX RCONST(1.0) /* Total range of x variable */
#define AY RCONST(1.0) /* Total range of y variable */
#define RTOL RCONST(1.e-5) /* Relative tolerance */
#define ATOL RCONST(1.e-5) /* Absolute tolerance */
#define NOUT 6 /* Number of output times */
#define TMULT RCONST(10.0) /* Multiplier for tout values */
#define TADD RCONST(0.3) /* Increment for tout values */
#define ZERO RCONST(0.)
#define ONE RCONST(1.0)
/*
* User-defined vector and accessor macro: IJ_Vptr.
* IJ_Vptr is defined in order to express the underlying 3-D structure of
* the dependent variable vector from its underlying 1-D storage (an N_Vector).
* IJ_Vptr(vv,i,j) returns a pointer to the location in vv corresponding to
* species index is = 0, x-index ix = i, and y-index jy = j.
*/
#define IJ_Vptr(vv,i,j) (&NV_Ith_S(vv, (i)*NUM_SPECIES + (j)*NSMX))
/* Type: UserData. Contains problem constants, etc. */
typedef struct {
long int Neq, ns, np, mx, my;
realtype dx, dy, **acoef;
realtype cox[NUM_SPECIES], coy[NUM_SPECIES], bcoef[NUM_SPECIES];
N_Vector rates;
} *UserData;
/* Prototypes for functions called by the IDA Solver. */
static int resweb(realtype time, N_Vector cc, N_Vector cp, N_Vector resval,
void *user_data);
/* Prototypes for private Helper Functions. */
static void InitUserData(UserData webdata);
static void SetInitialProfiles(N_Vector cc, N_Vector cp, N_Vector id,
UserData webdata);
static void PrintHeader(long int mu, long int ml, realtype rtol, realtype atol);
static void PrintOutput(void *mem, N_Vector c, realtype t);
static void PrintFinalStats(void *mem);
static void Fweb(realtype tcalc, N_Vector cc, N_Vector crate, UserData webdata);
static void WebRates(realtype xx, realtype yy, realtype *cxy, realtype *ratesxy,
UserData webdata);
static realtype dotprod(long int size, realtype *x1, realtype *x2);
static int check_flag(void *flagvalue, char *funcname, int opt);
/*
*--------------------------------------------------------------------
* MAIN PROGRAM
*--------------------------------------------------------------------
*/
int main()
{
void *mem;
UserData webdata;
N_Vector cc, cp, id;
int iout, retval;
long int mu, ml;
realtype rtol, atol, t0, tout, tret;
mem = NULL;
webdata = NULL;
cc = cp = id = NULL;
/* Allocate and initialize user data block webdata. */
webdata = (UserData) malloc(sizeof *webdata);
webdata->rates = N_VNew_Serial(NEQ);
webdata->acoef = newDenseMat(NUM_SPECIES, NUM_SPECIES);
InitUserData(webdata);
/* Allocate N-vectors and initialize cc, cp, and id. */
cc = N_VNew_Serial(NEQ);
if(check_flag((void *)cc, "N_VNew_Serial", 0)) return(1);
cp = N_VNew_Serial(NEQ);
if(check_flag((void *)cp, "N_VNew_Serial", 0)) return(1);
id = N_VNew_Serial(NEQ);
if(check_flag((void *)id, "N_VNew_Serial", 0)) return(1);
SetInitialProfiles(cc, cp, id, webdata);
/* Set remaining inputs to IDAMalloc. */
t0 = ZERO;
rtol = RTOL;
atol = ATOL;
/* Call IDACreate and IDAMalloc to initialize IDA. */
mem = IDACreate();
if(check_flag((void *)mem, "IDACreate", 0)) return(1);
retval = IDASetUserData(mem, webdata);
if(check_flag(&retval, "IDASetUserData", 1)) return(1);
retval = IDASetId(mem, id);
if(check_flag(&retval, "IDASetId", 1)) return(1);
retval = IDAInit(mem, resweb, t0, cc, cp);
if(check_flag(&retval, "IDAInit", 1)) return(1);
retval = IDASStolerances(mem, rtol, atol);
if(check_flag(&retval, "IDASStolerances", 1)) return(1);
/* Call IDABand to specify the IDA linear solver. */
mu = ml = NSMX;
retval = IDABand(mem, NEQ, mu, ml);
if(check_flag(&retval, "IDABand", 1)) return(1);
/* Call IDACalcIC (with default options) to correct the initial values. */
tout = RCONST(0.001);
retval = IDACalcIC(mem, IDA_YA_YDP_INIT, tout);
if(check_flag(&retval, "IDACalcIC", 1)) return(1);
/* Print heading, basic parameters, and initial values. */
PrintHeader(mu, ml, rtol, atol);
PrintOutput(mem, cc, ZERO);
/* Loop over iout, call IDASolve (normal mode), print selected output. */
for (iout = 1; iout <= NOUT; iout++) {
retval = IDASolve(mem, tout, &tret, cc, cp, IDA_NORMAL);
if(check_flag(&retval, "IDASolve", 1)) return(retval);
PrintOutput(mem, cc, tret);
if (iout < 3) tout *= TMULT; else tout += TADD;
}
/* Print final statistics and free memory. */
PrintFinalStats(mem);
/* Free memory */
IDAFree(&mem);
N_VDestroy_Serial(cc);
N_VDestroy_Serial(cp);
N_VDestroy_Serial(id);
destroyMat(webdata->acoef);
N_VDestroy_Serial(webdata->rates);
free(webdata);
return(0);
}
/* Define lines for readability in later routines */
#define acoef (webdata->acoef)
#define bcoef (webdata->bcoef)
#define cox (webdata->cox)
#define coy (webdata->coy)
/*
*--------------------------------------------------------------------
* FUNCTIONS CALLED BY IDA
*--------------------------------------------------------------------
*/
/*
* resweb: System residual function for predator-prey system.
* This routine calls Fweb to get all the right-hand sides of the
* equations, then loads the residual vector accordingly,
* using cp in the case of prey species.
*/
static int resweb(realtype tt, N_Vector cc, N_Vector cp,
N_Vector res, void *user_data)
{
long int jx, jy, is, yloc, loc, np;
realtype *resv, *cpv;
UserData webdata;
webdata = (UserData)user_data;
cpv = NV_DATA_S(cp);
resv = NV_DATA_S(res);
np = webdata->np;
/* Call Fweb to set res to vector of right-hand sides. */
Fweb(tt, cc, res, webdata);
/* Loop over all grid points, setting residual values appropriately
for differential or algebraic components. */
for (jy = 0; jy < MY; jy++) {
yloc = NSMX * jy;
for (jx = 0; jx < MX; jx++) {
loc = yloc + NUM_SPECIES * jx;
for (is = 0; is < NUM_SPECIES; is++) {
if (is < np)
resv[loc+is] = cpv[loc+is] - resv[loc+is];
else
resv[loc+is] = -resv[loc+is];
}
}
}
return(0);
}
/*
*--------------------------------------------------------------------
* PRIVATE FUNCTIONS
*--------------------------------------------------------------------
*/
/*
* InitUserData: Load problem constants in webdata (of type UserData).
*/
static void InitUserData(UserData webdata)
{
int i, j, np;
realtype *a1,*a2, *a3, *a4, dx2, dy2;
webdata->mx = MX;
webdata->my = MY;
webdata->ns = NUM_SPECIES;
webdata->np = NPREY;
webdata->dx = AX/(MX-1);
webdata->dy = AY/(MY-1);
webdata->Neq= NEQ;
/* Set up the coefficients a and b, and others found in the equations. */
np = webdata->np;
dx2 = (webdata->dx)*(webdata->dx); dy2 = (webdata->dy)*(webdata->dy);
for (i = 0; i < np; i++) {
a1 = &(acoef[i][np]);
a2 = &(acoef[i+np][0]);
a3 = &(acoef[i][0]);
a4 = &(acoef[i+np][np]);
/* Fill in the portion of acoef in the four quadrants, row by row. */
for (j = 0; j < np; j++) {
*a1++ = -GG;
*a2++ = EE;
*a3++ = ZERO;
*a4++ = ZERO;
}
/* Reset the diagonal elements of acoef to -AA. */
acoef[i][i] = -AA; acoef[i+np][i+np] = -AA;
/* Set coefficients for b and diffusion terms. */
bcoef[i] = BB; bcoef[i+np] = -BB;
cox[i] = DPREY/dx2; cox[i+np] = DPRED/dx2;
coy[i] = DPREY/dy2; coy[i+np] = DPRED/dy2;
}
}
/*
* SetInitialProfiles: Set initial conditions in cc, cp, and id.
* A polynomial profile is used for the prey cc values, and a constant
* (1.0e5) is loaded as the initial guess for the predator cc values.
* The id values are set to 1 for the prey and 0 for the predators.
* The prey cp values are set according to the given system, and
* the predator cp values are set to zero.
*/
static void SetInitialProfiles(N_Vector cc, N_Vector cp, N_Vector id,
UserData webdata)
{
long int loc, yloc, is, jx, jy, np;
realtype xx, yy, xyfactor, fac;
realtype *ccv, *cpv, *idv;
ccv = NV_DATA_S(cc);
cpv = NV_DATA_S(cp);
idv = NV_DATA_S(id);
np = webdata->np;
/* Loop over grid, load cc values and id values. */
for (jy = 0; jy < MY; jy++) {
yy = jy * webdata->dy;
yloc = NSMX * jy;
for (jx = 0; jx < MX; jx++) {
xx = jx * webdata->dx;
xyfactor = RCONST(16.0)*xx*(ONE-xx)*yy*(ONE-yy);
xyfactor *= xyfactor;
loc = yloc + NUM_SPECIES*jx;
fac = ONE + ALPHA * xx * yy + BETA * sin(FOURPI*xx) * sin(FOURPI*yy);
for (is = 0; is < NUM_SPECIES; is++) {
if (is < np) {
ccv[loc+is] = RCONST(10.0) + (realtype)(is+1) * xyfactor;
idv[loc+is] = ONE;
}
else {
ccv[loc+is] = RCONST(1.0e5);
idv[loc+is] = ZERO;
}
}
}
}
/* Set c' for the prey by calling the function Fweb. */
Fweb(ZERO, cc, cp, webdata);
/* Set c' for predators to 0. */
for (jy = 0; jy < MY; jy++) {
yloc = NSMX * jy;
for (jx = 0; jx < MX; jx++) {
loc = yloc + NUM_SPECIES * jx;
for (is = np; is < NUM_SPECIES; is++) {
cpv[loc+is] = ZERO;
}
}
}
}
/*
* Print first lines of output (problem description)
*/
static void PrintHeader(long int mu, long int ml, realtype rtol, realtype atol)
{
printf("\nidaFoodWeb_bnd: Predator-prey DAE serial example problem for IDA \n\n");
printf("Number of species ns: %d", NUM_SPECIES);
printf(" Mesh dimensions: %d x %d", MX, MY);
printf(" System size: %d\n", NEQ);
#if defined(SUNDIALS_EXTENDED_PRECISION)
printf("Tolerance parameters: rtol = %Lg atol = %Lg\n", rtol, atol);
#elif defined(SUNDIALS_DOUBLE_PRECISION)
printf("Tolerance parameters: rtol = %lg atol = %lg\n", rtol, atol);
#else
printf("Tolerance parameters: rtol = %g atol = %g\n", rtol, atol);
#endif
printf("Linear solver: IDABAND, Band parameters mu = %ld, ml = %ld\n",mu,ml);
printf("CalcIC called to correct initial predator concentrations.\n\n");
printf("-----------------------------------------------------------\n");
printf(" t bottom-left top-right");
printf(" | nst k h\n");
printf("-----------------------------------------------------------\n\n");
}
/*
* PrintOutput: Print output values at output time t = tt.
* Selected run statistics are printed. Then values of the concentrations
* are printed for the bottom left and top right grid points only.
*/
static void PrintOutput(void *mem, N_Vector c, realtype t)
{
int i, kused, flag;
long int nst;
realtype *c_bl, *c_tr, hused;
flag = IDAGetLastOrder(mem, &kused);
check_flag(&flag, "IDAGetLastOrder", 1);
flag = IDAGetNumSteps(mem, &nst);
check_flag(&flag, "IDAGetNumSteps", 1);
flag = IDAGetLastStep(mem, &hused);
check_flag(&flag, "IDAGetLastStep", 1);
c_bl = IJ_Vptr(c,0,0);
c_tr = IJ_Vptr(c,MX-1,MY-1);
#if defined(SUNDIALS_EXTENDED_PRECISION)
printf("%8.2Le %12.4Le %12.4Le | %3ld %1d %12.4Le\n",
t, c_bl[0], c_tr[1], nst, kused, hused);
for (i=1;i<NUM_SPECIES;i++)
printf(" %12.4Le %12.4Le |\n",c_bl[i],c_tr[i]);
#elif defined(SUNDIALS_DOUBLE_PRECISION)
printf("%8.2le %12.4le %12.4le | %3ld %1d %12.4le\n",
t, c_bl[0], c_tr[1], nst, kused, hused);
for (i=1;i<NUM_SPECIES;i++)
printf(" %12.4le %12.4le |\n",c_bl[i],c_tr[i]);
#else
printf("%8.2e %12.4e %12.4e | %3ld %1d %12.4e\n",
t, c_bl[0], c_tr[1], nst, kused, hused);
for (i=1;i<NUM_SPECIES;i++)
printf(" %12.4e %12.4e |\n",c_bl[i],c_tr[i]);
#endif
printf("\n");
}
/*
* PrintFinalStats: Print final run data contained in iopt.
*/
static void PrintFinalStats(void *mem)
{
long int nst, nre, nreLS, nni, nje, netf, ncfn;
int flag;
flag = IDAGetNumSteps(mem, &nst);
check_flag(&flag, "IDAGetNumSteps", 1);
flag = IDAGetNumNonlinSolvIters(mem, &nni);
check_flag(&flag, "IDAGetNumNonlinSolvIters", 1);
flag = IDAGetNumResEvals(mem, &nre);
check_flag(&flag, "IDAGetNumResEvals", 1);
flag = IDAGetNumErrTestFails(mem, &netf);
check_flag(&flag, "IDAGetNumErrTestFails", 1);
flag = IDAGetNumNonlinSolvConvFails(mem, &ncfn);
check_flag(&flag, "IDAGetNumNonlinSolvConvFails", 1);
flag = IDADlsGetNumJacEvals(mem, &nje);
check_flag(&flag, "IDADlsGetNumJacEvals", 1);
flag = IDADlsGetNumResEvals(mem, &nreLS);
check_flag(&flag, "IDADlsGetNumResEvals", 1);
printf("-----------------------------------------------------------\n");
printf("Final run statistics: \n\n");
printf("Number of steps = %ld\n", nst);
printf("Number of residual evaluations = %ld\n", nre+nreLS);
printf("Number of Jacobian evaluations = %ld\n", nje);
printf("Number of nonlinear iterations = %ld\n", nni);
printf("Number of error test failures = %ld\n", netf);
printf("Number of nonlinear conv. failures = %ld\n", ncfn);
}
/*
* Fweb: Rate function for the food-web problem.
* This routine computes the right-hand sides of the system equations,
* consisting of the diffusion term and interaction term.
* The interaction term is computed by the function WebRates.
*/
static void Fweb(realtype tcalc, N_Vector cc, N_Vector crate,
UserData webdata)
{
long int jx, jy, is, idyu, idyl, idxu, idxl;
realtype xx, yy, *cxy, *ratesxy, *cratexy, dcyli, dcyui, dcxli, dcxui;
/* Loop over grid points, evaluate interaction vector (length ns),
form diffusion difference terms, and load crate. */
for (jy = 0; jy < MY; jy++) {
yy = (webdata->dy) * jy ;
idyu = (jy!=MY-1) ? NSMX : -NSMX;
idyl = (jy!= 0 ) ? NSMX : -NSMX;
for (jx = 0; jx < MX; jx++) {
xx = (webdata->dx) * jx;
idxu = (jx!= MX-1) ? NUM_SPECIES : -NUM_SPECIES;
idxl = (jx!= 0 ) ? NUM_SPECIES : -NUM_SPECIES;
cxy = IJ_Vptr(cc,jx,jy);
ratesxy = IJ_Vptr(webdata->rates,jx,jy);
cratexy = IJ_Vptr(crate,jx,jy);
/* Get interaction vector at this grid point. */
WebRates(xx, yy, cxy, ratesxy, webdata);
/* Loop over species, do differencing, load crate segment. */
for (is = 0; is < NUM_SPECIES; is++) {
/* Differencing in y. */
dcyli = *(cxy+is) - *(cxy - idyl + is) ;
dcyui = *(cxy + idyu + is) - *(cxy+is);
/* Differencing in x. */
dcxli = *(cxy+is) - *(cxy - idxl + is);
dcxui = *(cxy + idxu +is) - *(cxy+is);
/* Compute the crate values at (xx,yy). */
cratexy[is] = coy[is] * (dcyui - dcyli) +
cox[is] * (dcxui - dcxli) + ratesxy[is];
} /* End is loop */
} /* End of jx loop */
} /* End of jy loop */
}
/*
* WebRates: Evaluate reaction rates at a given spatial point.
* At a given (x,y), evaluate the array of ns reaction terms R.
*/
static void WebRates(realtype xx, realtype yy, realtype *cxy, realtype *ratesxy,
UserData webdata)
{
int is;
realtype fac;
for (is = 0; is < NUM_SPECIES; is++)
ratesxy[is] = dotprod(NUM_SPECIES, cxy, acoef[is]);
fac = ONE + ALPHA*xx*yy + BETA*sin(FOURPI*xx)*sin(FOURPI*yy);
for (is = 0; is < NUM_SPECIES; is++)
ratesxy[is] = cxy[is]*( bcoef[is]*fac + ratesxy[is] );
}
/*
* dotprod: dot product routine for realtype arrays, for use by WebRates.
*/
static realtype dotprod(long int size, realtype *x1, realtype *x2)
{
long int i;
realtype *xx1, *xx2, temp = ZERO;
xx1 = x1; xx2 = x2;
for (i = 0; i < size; i++) temp += (*xx1++) * (*xx2++);
return(temp);
}
/*
* 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;
if (opt == 0 && flagvalue == NULL) {
/* Check if SUNDIALS function returned NULL pointer - no memory allocated */
fprintf(stderr,
"\nSUNDIALS_ERROR: %s() failed - returned NULL pointer\n\n",
funcname);
return(1);
} else if (opt == 1) {
/* Check if flag < 0 */
errflag = (int *) flagvalue;
if (*errflag < 0) {
fprintf(stderr,
"\nSUNDIALS_ERROR: %s() failed with flag = %d\n\n",
funcname, *errflag);
return(1);
}
} else if (opt == 2 && flagvalue == NULL) {
/* Check if function returned NULL pointer - no memory allocated */
fprintf(stderr,
"\nMEMORY_ERROR: %s() failed - returned NULL pointer\n\n",
funcname);
return(1);
}
return(0);
}
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