Rheolef  7.2
an efficient C++ finite element environment
 
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burgers_diffusion_dg.cc

The diffusive Burgers equation by the discontinuous Galerkin method.

The diffusive Burgers equation by the discontinuous Galerkin method

#include "rheolef.h"
using namespace rheolef;
using namespace std;
#include "burgers.icc"
#undef NEUMANN
int main(int argc, char**argv) {
environment rheolef (argc, argv);
geo omega (argv[1]);
space Xh (omega, argv[2]);
size_t k = Xh.degree();
Float epsilon = (argc > 3) ? atof(argv[3]) : 0.1;
size_t nmax = (argc > 4) ? atoi(argv[4]) : 500;
Float tf = (argc > 5) ? atof(argv[5]) : 1;
size_t p = (argc > 6) ? atoi(argv[6]) : min(k+1,rk::pmax);
Float delta_t = tf/nmax;
size_t d = omega.dimension();
Float beta = (k+1)*(k+d)/Float(d);
trial u (Xh); test v (Xh);
form m = integrate (u*v);
iopt.invert = true;
form inv_m = integrate (u*v, iopt);
form a = epsilon*(
integrate (dot(grad_h(u),grad_h(v)))
#ifdef NEUMANN
+ integrate ("internal_sides",
#else // NEUMANN
+ integrate ("sides",
#endif // NEUMANN
beta*penalty()*jump(u)*jump(v)
- jump(u)*average(dot(grad_h(v),normal()))
- jump(v)*average(dot(grad_h(u),normal()))));
vector<problem> pb (p+1);
for (size_t i = 1; i <= p; ++i) {
form ci = m + delta_t*rk::alpha[p][i][i]*a;
pb[i] = problem(ci);
}
vector<field> uh(p+1, field(Xh,0));
uh[0] = lazy_interpolate (Xh, u_init(epsilon));
branch even("t","u");
dout << catchmark("epsilon") << epsilon << endl
<< even(0,uh[0]);
for (size_t n = 0; n < nmax; ++n) {
Float tn = n*delta_t;
Float t = tn + delta_t;
field uh_next = uh[0] - delta_t*rk::tilde_beta[p][0]*(inv_m*gh(epsilon, tn, uh[0], v));
for (size_t i = 1; i <= p; ++i) {
Float ti = tn + rk::gamma[p][i]*delta_t;
field rhs = m*uh[0] - delta_t*rk::tilde_alpha[p][i][0]*gh(epsilon, tn, uh[0], v);
for (size_t j = 1; j <= i-1; ++j) {
Float tj = tn + rk::gamma[p][j]*delta_t;
rhs -= delta_t*( rk::alpha[p][i][j]*(a*uh[j] - lh(epsilon,tj,v))
+ rk::tilde_alpha[p][i][j]*gh(epsilon, tj, uh[j], v));
}
rhs += delta_t*rk::alpha[p][i][i]*lh (epsilon, ti, v);
pb[i].solve (rhs, uh[i]);
uh_next -= delta_t*(inv_m*( rk::beta[p][i]*(a*uh[i] - lh(epsilon,ti,v))
+ rk::tilde_beta[p][i]*gh(epsilon, ti, uh[i], v)));
}
uh_next = limiter(uh_next);
dout << even(tn+delta_t,uh_next);
uh[0] = uh_next;
}
}
The Burgers equation – the f function.
The diffusive Burgers equation – its exact solution.
u_exact u_init
The diffusive Burgers equation – operators.
field lh(Float epsilon, Float t, const test &v)
field gh(Float epsilon, Float t, const field &uh, const test &v)
The Burgers equation – the Godonov flux.
see the Float page for the full documentation
see the branch page for the full documentation
see the field page for the full documentation
see the form page for the full documentation
see the geo page for the full documentation
see the problem page for the full documentation
see the catchmark page for the full documentation
Definition catchmark.h:67
see the environment page for the full documentation
see the integrate_option page for the full documentation
see the space page for the full documentation
see the test page for the full documentation
see the test page for the full documentation
int main()
Definition field2bb.cc:58
This file is part of Rheolef.
Float tilde_alpha[][pmax+1][pmax+1]
Float tilde_beta[][pmax+1]
Float gamma[][pmax+1]
Float beta[][pmax+1]
Float alpha[][pmax+1][pmax+1]
constexpr size_t pmax
STL namespace.
rheolef - reference manual
The semi-implicit Runge-Kutta scheme – coefficients.
Definition sphere.icc:25
Definition leveque.h:25
Float epsilon