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max_delta and adaptive norm in rkf45 and rkbs32

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This commit is contained in:
Ian Jauslin 2023-06-14 00:14:17 -04:00
parent 251426aaaa
commit aeec3ff17b
2 changed files with 112 additions and 32 deletions
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140
src/navier-stokes.c
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@ -92,13 +92,13 @@ int uk(
} else if (algorithm==ALGORITHM_RK4) {
ns_step_rk4(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, irreversible);
} else if (algorithm==ALGORITHM_RKF45) {
ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, true);
ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, true);
} else if (algorithm==ALGORITHM_RKDP54) {
// only compute k1 on the first step
ns_step_rkdp54(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, time==starting_time);
} else if (algorithm==ALGORITHM_RKBS32) {
// only compute k1 on the first step
ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, time==starting_time);
ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, time==starting_time);
} else {
fprintf(stderr,"bug: unknown algorithm: %u, contact ian.jauslin@rutgers,edu\n",algorithm);
}
@ -203,13 +203,13 @@ int enstrophy(
} else if (algorithm==ALGORITHM_RK4) {
ns_step_rk4(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, irreversible);
} else if (algorithm==ALGORITHM_RKF45) {
ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, true);
ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, true);
} else if (algorithm==ALGORITHM_RKDP54) {
// only compute k1 on the first step
ns_step_rkdp54(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, time==starting_time);
} else if (algorithm==ALGORITHM_RKBS32) {
// only compute k1 on the first step
ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, time==starting_time);
ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, time==starting_time);
} else {
fprintf(stderr,"bug: unknown algorithm: %u, contact ian.jauslin@rutgers,edu\n",algorithm);
}
@ -363,13 +363,13 @@ int quiet(
} else if (algorithm==ALGORITHM_RK4) {
ns_step_rk4(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, irreversible);
} else if (algorithm==ALGORITHM_RKF45) {
ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, true);
ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, true);
} else if (algorithm==ALGORITHM_RKDP54) {
// only compute k1 on the first step
ns_step_rkdp54(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, time==starting_time);
} else if (algorithm==ALGORITHM_RKBS32) {
// only compute k1 on the first step
ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, time==starting_time);
ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, time==starting_time);
} else {
fprintf(stderr,"bug: unknown algorithm: %u, contact ian.jauslin@rutgers,edu\n",algorithm);
}
@ -649,6 +649,8 @@ int ns_step_rkf45(
_Complex double* u,
double tolerance,
double factor,
double max_delta,
unsigned int adaptive_norm,
int K1,
int K2,
int N1,
@ -722,17 +724,59 @@ int ns_step_rkf45(
// compute error
err=0;
relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
// difference between 5th order and 4th order
// use the norm |u_i|/k^3
err+=cabs((*delta)*(1./360*k1[klookup_sym(kx,ky,K2)]-128./4275*k3[klookup_sym(kx,ky,K2)]-2197./75240*k4[klookup_sym(kx,ky,K2)]+1./50*k5[klookup_sym(kx,ky,K2)]+2./55*k6[klookup_sym(kx,ky,K2)]))/pow(kx*kx+ky*ky,1.5);
// next step
tmp[klookup_sym(kx,ky,K2)]=(*delta)*(25./216*k1[klookup_sym(kx,ky,K2)]+1408./2565*k3[klookup_sym(kx,ky,K2)]+2197./4104*k4[klookup_sym(kx,ky,K2)]-1./5*k5[klookup_sym(kx,ky,K2)]);
relative+=cabs(tmp[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,1.5);
if(adaptive_norm==NORM_L1){
relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=cabs((*delta)*(1./360*k1[klookup_sym(kx,ky,K2)]-128./4275*k3[klookup_sym(kx,ky,K2)]-2197./75240*k4[klookup_sym(kx,ky,K2)]+1./50*k5[klookup_sym(kx,ky,K2)]+2./55*k6[klookup_sym(kx,ky,K2)]));
// next step
tmp[klookup_sym(kx,ky,K2)]=(*delta)*(25./216*k1[klookup_sym(kx,ky,K2)]+1408./2565*k3[klookup_sym(kx,ky,K2)]+2197./4104*k4[klookup_sym(kx,ky,K2)]-1./5*k5[klookup_sym(kx,ky,K2)]);
relative+=cabs(tmp[klookup_sym(kx,ky,K2)]);
}
}
}
else if(adaptive_norm==NORM_k3){
relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=cabs((*delta)*(1./360*k1[klookup_sym(kx,ky,K2)]-128./4275*k3[klookup_sym(kx,ky,K2)]-2197./75240*k4[klookup_sym(kx,ky,K2)]+1./50*k5[klookup_sym(kx,ky,K2)]+2./55*k6[klookup_sym(kx,ky,K2)]))/pow(kx*kx+ky*ky,1.5);
// next step
tmp[klookup_sym(kx,ky,K2)]=(*delta)*(25./216*k1[klookup_sym(kx,ky,K2)]+1408./2565*k3[klookup_sym(kx,ky,K2)]+2197./4104*k4[klookup_sym(kx,ky,K2)]-1./5*k5[klookup_sym(kx,ky,K2)]);
relative+=cabs(tmp[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,1.5);
}
}
}
else if(adaptive_norm==NORM_k32){
relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=cabs((*delta)*(1./360*k1[klookup_sym(kx,ky,K2)]-128./4275*k3[klookup_sym(kx,ky,K2)]-2197./75240*k4[klookup_sym(kx,ky,K2)]+1./50*k5[klookup_sym(kx,ky,K2)]+2./55*k6[klookup_sym(kx,ky,K2)]))/pow(kx*kx+ky*ky,0.75);
// next step
tmp[klookup_sym(kx,ky,K2)]=(*delta)*(25./216*k1[klookup_sym(kx,ky,K2)]+1408./2565*k3[klookup_sym(kx,ky,K2)]+2197./4104*k4[klookup_sym(kx,ky,K2)]-1./5*k5[klookup_sym(kx,ky,K2)]);
relative+=cabs(tmp[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,0.75);
}
}
}
else if(adaptive_norm==NORM_ENSTROPHY){
double sumu, sumU;
sumu=0;
sumU=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=(kx*kx+ky*ky)*cabs2((*delta)*(1./360*k1[klookup_sym(kx,ky,K2)]-128./4275*k3[klookup_sym(kx,ky,K2)]-2197./75240*k4[klookup_sym(kx,ky,K2)]+1./50*k5[klookup_sym(kx,ky,K2)]+2./55*k6[klookup_sym(kx,ky,K2)]));
// next step
tmp[klookup_sym(kx,ky,K2)]=(*delta)*(25./216*k1[klookup_sym(kx,ky,K2)]+1408./2565*k3[klookup_sym(kx,ky,K2)]+2197./4104*k4[klookup_sym(kx,ky,K2)]-1./5*k5[klookup_sym(kx,ky,K2)]);
sumU+=(kx*kx+ky*ky)*cabs2((*delta)*(16./135*k1[klookup_sym(kx,ky,K2)]+6656./12825*k3[klookup_sym(kx,ky,K2)]+28561./56430*k4[klookup_sym(kx,ky,K2)]-9./50*k5[klookup_sym(kx,ky,K2)]+2./55*k6[klookup_sym(kx,ky,K2)]));
sumu+=(kx*kx+ky*ky)*cabs2(u[klookup_sym(kx,ky,K2)]+tmp[klookup_sym(kx,ky,K2)]);
}
}
err=sqrt(err);
relative=pow((sqrt(sumu)+sqrt(sumU))/sumu, 1./3);
}
else{
fprintf(stderr,"bug: unknown norm: %u, contact ian.jauslin@rutgers.edu\n", adaptive_norm);
exit(-1);
}
// compare relative error with tolerance
if(err<relative*tolerance){
@ -742,14 +786,14 @@ int ns_step_rkf45(
u[klookup_sym(kx,ky,K2)]+=tmp[klookup_sym(kx,ky,K2)];
}
}
// next delta to use in future steps
*next_delta=(*delta)*pow(relative*tolerance/err,0.2);
// next delta to use in future steps (do not exceed max_delta)
*next_delta=fmin(max_delta, (*delta)*pow(relative*tolerance/err,0.2));
}
// error too big: repeat with smaller step
else{
*delta=factor*(*delta)*pow(relative*tolerance/err,0.2);
// do not recompute k1
ns_step_rkf45(u,tolerance,factor,K1,K2,N1,N2,nu,delta,next_delta,L,g,fft1,fft2,ifft,k1,k2,k3,k4,k5,k6,tmp,irreversible,false);
ns_step_rkf45(u,tolerance,factor,max_delta,adaptive_norm,K1,K2,N1,N2,nu,delta,next_delta,L,g,fft1,fft2,ifft,k1,k2,k3,k4,k5,k6,tmp,irreversible,false);
}
return 0;
@ -762,6 +806,8 @@ int ns_step_rkbs32(
_Complex double* u,
double tolerance,
double factor,
double max_delta,
unsigned int adaptive_norm,
int K1,
int K2,
int N1,
@ -820,14 +866,51 @@ int ns_step_rkbs32(
// compute error
err=0;
relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
// difference between 5th order and 4th order
err+=cabs((*delta)*(5./72*(*k1)[klookup_sym(kx,ky,K2)]-1./12*k2[klookup_sym(kx,ky,K2)]-1./9*k3[klookup_sym(kx,ky,K2)]+1./8*(*k4)[klookup_sym(kx,ky,K2)]))/pow(kx*kx+ky*ky,0.75);
relative+=cabs(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,0.75);
if(adaptive_norm==NORM_L1){
relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=cabs((*delta)*(5./72*(*k1)[klookup_sym(kx,ky,K2)]-1./12*k2[klookup_sym(kx,ky,K2)]-1./9*k3[klookup_sym(kx,ky,K2)]+1./8*(*k4)[klookup_sym(kx,ky,K2)]));
relative+=cabs(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)]);
}
}
}
else if(adaptive_norm==NORM_k3){
relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=cabs((*delta)*(5./72*(*k1)[klookup_sym(kx,ky,K2)]-1./12*k2[klookup_sym(kx,ky,K2)]-1./9*k3[klookup_sym(kx,ky,K2)]+1./8*(*k4)[klookup_sym(kx,ky,K2)]))/pow(kx*kx+ky*ky,1.5);
relative+=cabs(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,1.5);
}
}
}
else if(adaptive_norm==NORM_k32){
relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=cabs((*delta)*(5./72*(*k1)[klookup_sym(kx,ky,K2)]-1./12*k2[klookup_sym(kx,ky,K2)]-1./9*k3[klookup_sym(kx,ky,K2)]+1./8*(*k4)[klookup_sym(kx,ky,K2)]))/pow(kx*kx+ky*ky,0.75);
relative+=cabs(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,0.75);
}
}
}
else if(adaptive_norm==NORM_ENSTROPHY){
double sumu, sumU;
sumu=0;
sumU=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=(kx*kx+ky*ky)*cabs2((*delta)*(5./72*(*k1)[klookup_sym(kx,ky,K2)]-1./12*k2[klookup_sym(kx,ky,K2)]-1./9*k3[klookup_sym(kx,ky,K2)]+1./8*(*k4)[klookup_sym(kx,ky,K2)]));
sumU+=(kx*kx+ky*ky)*cabs2((*delta)*(7./24*(*k1)[klookup_sym(kx,ky,K2)]+1./4*k2[klookup_sym(kx,ky,K2)]+1./3*k3[klookup_sym(kx,ky,K2)]+1./8*(*k4)[klookup_sym(kx,ky,K2)]));
sumu+=(kx*kx+ky*ky)*cabs2(tmp[klookup_sym(kx,ky,K2)]);
}
}
err=sqrt(err);
relative=pow((sqrt(sumu)+sqrt(sumU))/sumu, 1./3);
}
else{
fprintf(stderr,"bug: unknown norm: %u, contact ian.jauslin@rutgers,edu\n", adaptive_norm);
exit(-1);
}
// compare relative error with tolerance
if(err<relative*tolerance){
@ -837,8 +920,8 @@ int ns_step_rkbs32(
u[klookup_sym(kx,ky,K2)]=tmp[klookup_sym(kx,ky,K2)];
}
}
// next delta to use in future steps
*next_delta=(*delta)*pow(relative*tolerance/err,1./3);
// next delta to use in future steps (do not exceed max_delta)
*next_delta=fmin(max_delta, (*delta)*pow(relative*tolerance/err,1./3));
// k1 in the next step will be this k4 (first same as last)
tmp=*k1;
@ -849,7 +932,7 @@ int ns_step_rkbs32(
else{
*delta=factor*(*delta)*pow(relative*tolerance/err,1./3);
// this will reuse the same k1 without re-computing it
ns_step_rkbs32(u,tolerance,factor,K1,K2,N1,N2,nu,delta,next_delta,L,g,fft1,fft2,ifft,k1,k2,k3,k4,tmp,irreversible,false);
ns_step_rkbs32(u,tolerance,factor,max_delta,adaptive_norm,K1,K2,N1,N2,nu,delta,next_delta,L,g,fft1,fft2,ifft,k1,k2,k3,k4,tmp,irreversible,false);
}
return 0;
@ -948,7 +1031,6 @@ int ns_step_rkdp54(
// compute error
err=0;
// norm: |u_k|
if(adaptive_norm==NORM_L1){
relative=0;
for(kx=0;kx<=K1;kx++){
@ -982,8 +1064,6 @@ int ns_step_rkdp54(
sumU=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
// difference between 5th order and 4th order
// use the norm |u_k|^2k^2 (to get a bound on the error of the enstrophy)
err+=(kx*kx+ky*ky)*cabs2((*delta)*(-71./57600*(*k1)[klookup_sym(kx,ky,K2)]+71./16695*k3[klookup_sym(kx,ky,K2)]-71./1920*k4[klookup_sym(kx,ky,K2)]+17253./339200*k5[klookup_sym(kx,ky,K2)]-22./525*k6[klookup_sym(kx,ky,K2)]+1./40*(*k2)[klookup_sym(kx,ky,K2)]));
sumU+=(kx*kx+ky*ky)*cabs2(u[klookup_sym(kx,ky,K2)]+(*delta)*(5179./57600*(*k1)[klookup_sym(kx,ky,K2)]+7571./16695*k3[klookup_sym(kx,ky,K2)]+393./640*k4[klookup_sym(kx,ky,K2)]-92097./339200*k5[klookup_sym(kx,ky,K2)]+187./2100*k6[klookup_sym(kx,ky,K2)]+1./40*(*k2)[klookup_sym(kx,ky,K2)]));
sumu+=(kx*kx+ky*ky)*cabs2(tmp[klookup_sym(kx,ky,K2)]);

4
src/navier-stokes.h
View File

@ -56,11 +56,11 @@ int ns_step_rk4( _Complex double* u, int K1, int K2, int N1, int N2, double nu,
// RK2
int ns_step_rk2( _Complex double* u, int K1, int K2, int N1, int N2, double nu, double delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2,fft_vect ifft, _Complex double* tmp1, _Complex double *tmp2, bool irreversible);
// Runge-Kutta-Fehlberg
int ns_step_rkf45( _Complex double* u, double tolerance, double factor, int K1, int K2, int N1, int N2, double nu, double* delta, double* next_delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double* k1, _Complex double* k2, _Complex double* k3, _Complex double* k4, _Complex double* k5, _Complex double* k6, _Complex double* tmp, bool irreversible, bool compute_k1);
int ns_step_rkf45( _Complex double* u, double tolerance, double factor, double max_delta, unsigned int adaptive_norm, int K1, int K2, int N1, int N2, double nu, double* delta, double* next_delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double* k1, _Complex double* k2, _Complex double* k3, _Complex double* k4, _Complex double* k5, _Complex double* k6, _Complex double* tmp, bool irreversible, bool compute_k1);
// Runge-Kutta-Dromand-Prince
int ns_step_rkdp54( _Complex double* u, double tolerance, double factor, double max_delta, unsigned int adaptive_norm, int K1, int K2, int N1, int N2, double nu, double* delta, double* next_delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double** k1, _Complex double** k2, _Complex double* k3, _Complex double* k4, _Complex double* k5, _Complex double* k6, _Complex double* tmp, bool irreversible, bool compute_k1);
// Runge-Kutta-Bogacki-Shampine
int ns_step_rkbs32( _Complex double* u, double tolerance, double factor, int K1, int K2, int N1, int N2, double nu, double* delta, double* next_delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double** k1, _Complex double* k2, _Complex double* k3, _Complex double** k4, _Complex double* tmp, bool irreversible, bool compute_k1);
int ns_step_rkbs32( _Complex double* u, double tolerance, double factor, double max_delta, unsigned int adaptive_norm, int K1, int K2, int N1, int N2, double nu, double* delta, double* next_delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double** k1, _Complex double* k2, _Complex double* k3, _Complex double** k4, _Complex double* tmp, bool irreversible, bool compute_k1);
// right side of Irreversible/reversible Navier-Stokes equation
int ns_rhs( _Complex double* out, _Complex double* u, int K1, int K2, int N1, int N2, double nu, double L, _Complex double* g, fft_vect fft1, fft_vect fft2, fft_vect ifft, bool irreversible);