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RKDP54

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This commit is contained in:
Ian Jauslin
2023-05-17 17:41:00 -04:00
parent dffa378084
commit 051cd84a29
5 changed files with 153 additions and 7 deletions
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8
README.md
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@ -93,9 +93,11 @@ should be a `;` sperated list of `key=value` pairs. The possible keys are
* `random_seed` (int, default ): seed for random initialization.
* `algorithm`: `RK4` for Runge-Kutta 4, `RK2` for Runge-Kutta 2, `RKF45` for
the Runge-Kutta-Fehlberg adaptive step method, `RKBS23` for the
Runge-Kutta-Bogacki-Shampine method.
* `algorithm`: fixed step methods: `RK4` for Runge-Kutta 4, `RK2` for
Runge-Kutta 2.
adaptive step methods: `RKF45` for Runge-Kutta-Fehlberg (order
4), `RKDP54` for Runge-Kutta-Dormand-Prince (order 5), `RKBS32` for
Runge-Kutta-Bogacki-Shampine (order 3).
* `adaptive_tolerance` (double, default 1e-11): when using an adaptive step
method, this is the maximal allowed relative error.

3
src/constants.cpp
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@ -34,5 +34,6 @@ limitations under the License.
#define ALGORITHM_ADAPTIVE_THRESHOLD 1000
// adaptive algorithms: index > ALGORITHM_ADAPTIVE_THRESHOLD
#define ALGORITHM_RKF45 1001
#define ALGORITHM_RKBS32 1002
#define ALGORITHM_RKDP54 1002
#define ALGORITHM_RKBS32 1003

6
src/main.c
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@ -265,6 +265,9 @@ int print_params(
case ALGORITHM_RKF45:
fprintf(file,", algorithm=RKF45, tolerance=%.15e, factor=%.15e",parameters.adaptive_tolerance, parameters.adaptive_factor);
break;
case ALGORITHM_RKDP54:
fprintf(file,", algorithm=RKDP54, tolerance=%.15e, factor=%.15e",parameters.adaptive_tolerance, parameters.adaptive_factor);
break;
case ALGORITHM_RKBS32:
fprintf(file,", algorithm=RKBS32, tolerance=%.15e, factor=%.15e",parameters.adaptive_tolerance, parameters.adaptive_factor);
break;
@ -674,6 +677,9 @@ int set_parameter(
else if (strcmp(rhs,"RKF45")==0){
parameters->algorithm=ALGORITHM_RKF45;
}
else if (strcmp(rhs,"RKDP54")==0){
parameters->algorithm=ALGORITHM_RKDP54;
}
else if (strcmp(rhs,"RKBS32")==0){
parameters->algorithm=ALGORITHM_RKBS32;
}

139
src/navier-stokes.c
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@ -90,6 +90,9 @@ int uk(
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);
} else if (algorithm==ALGORITHM_RKDP54) {
// only compute k1 on the first step
ns_step_rkdp54(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, 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);
@ -196,6 +199,9 @@ int enstrophy(
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);
} else if (algorithm==ALGORITHM_RKDP54) {
// only compute k1 on the first step
ns_step_rkdp54(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, 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);
@ -351,6 +357,9 @@ int quiet(
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);
} else if (algorithm==ALGORITHM_RKDP54) {
// only compute k1 on the first step
ns_step_rkdp54(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, 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);
@ -401,7 +410,7 @@ int ns_init_tmps(
*tmp1=calloc(sizeof(_Complex double),K1*(2*K2+1)+K2);
*tmp2=calloc(sizeof(_Complex double),K1*(2*K2+1)+K2);
*tmp3=calloc(sizeof(_Complex double),K1*(2*K2+1)+K2);
} else if (algorithm==ALGORITHM_RKF45){
} else if (algorithm==ALGORITHM_RKF45 || algorithm==ALGORITHM_RKDP54){
*tmp1=calloc(sizeof(_Complex double),K1*(2*K2+1)+K2);
*tmp2=calloc(sizeof(_Complex double),K1*(2*K2+1)+K2);
*tmp3=calloc(sizeof(_Complex double),K1*(2*K2+1)+K2);
@ -466,7 +475,7 @@ int ns_free_tmps(
free(tmp1);
free(tmp2);
free(tmp3);
} else if (algorithm==ALGORITHM_RKF45){
} else if (algorithm==ALGORITHM_RKF45 || algorithm==ALGORITHM_RKDP54){
free(tmp1);
free(tmp2);
free(tmp3);
@ -838,6 +847,132 @@ int ns_step_rkbs32(
return 0;
}
// next time step
// adaptive RK algorithm (Runge-Kutta-Dormand-Prince method)
int ns_step_rkdp54(
_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,
// the pointers k1 and k2 will be exchanged at the end of the routine
_Complex double** k1,
_Complex double** k2,
_Complex double* k3,
_Complex double* k4,
_Complex double* k5,
_Complex double* k6,
_Complex double* tmp,
bool irreversible,
// whether to compute k1
bool compute_k1
){
int kx,ky;
double err,relative;
// k1: u(t)
// only compute it if it is the first step (otherwise, it has already been computed due to the First Same As Last property)
if(compute_k1){
ns_rhs(*k1, u, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible);
}
// k2 : u(t+1/5*delta)
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
tmp[klookup_sym(kx,ky,K2)]=u[klookup_sym(kx,ky,K2)]+(*delta)/5*(*k1)[klookup_sym(kx,ky,K2)];
}
}
ns_rhs(*k2, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible);
// k3 : u(t+3/10*delta)
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
tmp[klookup_sym(kx,ky,K2)]=u[klookup_sym(kx,ky,K2)]+(*delta)*(3./40*(*k1)[klookup_sym(kx,ky,K2)]+9./40*(*k2)[klookup_sym(kx,ky,K2)]);
}
}
ns_rhs(k3, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible);
// k4 : u(t+4/5*delta)
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
tmp[klookup_sym(kx,ky,K2)]=u[klookup_sym(kx,ky,K2)]+(*delta)*(44./45*(*k1)[klookup_sym(kx,ky,K2)]-56./15*(*k2)[klookup_sym(kx,ky,K2)]+32./9*k3[klookup_sym(kx,ky,K2)]);
}
}
ns_rhs(k4, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible);
// k5 : u(t+8/9*delta)
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
tmp[klookup_sym(kx,ky,K2)]=u[klookup_sym(kx,ky,K2)]+(*delta)*(19372./6561*(*k1)[klookup_sym(kx,ky,K2)]-25360./2187*(*k2)[klookup_sym(kx,ky,K2)]+64448./6561*k3[klookup_sym(kx,ky,K2)]-212./729*k4[klookup_sym(kx,ky,K2)]);
}
}
ns_rhs(k5, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible);
// k6 : u(t+delta)
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
tmp[klookup_sym(kx,ky,K2)]=u[klookup_sym(kx,ky,K2)]+(*delta)*(9017./3168*(*k1)[klookup_sym(kx,ky,K2)]-355./33*(*k2)[klookup_sym(kx,ky,K2)]+46732./5247*k3[klookup_sym(kx,ky,K2)]+49./176*k4[klookup_sym(kx,ky,K2)]-5103./18656*k5[klookup_sym(kx,ky,K2)]);
}
}
ns_rhs(k6, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible);
// k7 : u(t+delta)
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
// tmp computed here is the step
tmp[klookup_sym(kx,ky,K2)]=u[klookup_sym(kx,ky,K2)]+(*delta)*(35./384*(*k1)[klookup_sym(kx,ky,K2)]+500./1113*k3[klookup_sym(kx,ky,K2)]+125./192*k4[klookup_sym(kx,ky,K2)]-2187./6784*k5[klookup_sym(kx,ky,K2)]+11./84*k6[klookup_sym(kx,ky,K2)]);
}
}
// store in k2, which is not needed anymore
ns_rhs(*k2, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible);
// 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)*(-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)]));
relative+=cabs(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)]);
}
}
// compare relative error with tolerance
if(err<relative*tolerance){
// add to output
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
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./5);
// k1 in the next step will be this k4 (first same as last)
tmp=*k1;
*k1=*k2;
*k2=tmp;
}
// error too big: repeat with smaller step
else{
*delta=factor*(*delta)*pow(relative*tolerance/err,1./5);
// this will reuse the same k1 without re-computing it
ns_step_rkdp54(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);
}
return 0;
}
// right side of Irreversible/Reversible Navier-Stokes equation
int ns_rhs(
_Complex double* out,

4
src/navier-stokes.h
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@ -55,8 +55,10 @@ int copy_u( _Complex double* u, _Complex double* u0, int K1, int K2);
int ns_step_rk4( _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, _Complex double *tmp3, bool irreversible);
// 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);
// adaptive RK algorithm (Runge-Kutta-Fehlberg)
// 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);
// Runge-Kutta-Dromand-Prince
int ns_step_rkdp54( _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);
// 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);