I tried to implement a solver for the non-linear diffusion equation
(\partial_t u = grad(u(grad u)) - f) using the TimeStepping-Class, the
EmbeddedExplicitRungeKutta-Method and (for assembly) the matrix-free
approach. For initial tests I used the linear heat equation with the
solution u = exp(-pi * pi * t) * sin(pi * x) * sin(pi * y) and the assembly
routine
template <int dim, int degree, int n_points_1d>
void LaplaceOperator<dim, degree, n_points_1d>::local_apply_cell(
const MatrixFree<dim, Number> & data,
vector_t & dst,
const vector_t &src,
const std::pair<unsigned int, unsigned int> & cell_range)
const
{
FEEvaluation<dim, degree, n_points_1d, n_components_to_use, Number>
phi(data);
for (unsigned int cell = cell_range.first; cell < cell_range.second;
++cell)
{
phi.reinit(cell);
phi.read_dof_values_plain(src);
phi.evaluate(false, true);
for (unsigned int q = 0; q < phi.n_q_points; ++q)
{
auto gradient_coefficient = calculate_gradient_coefficient(
phi.get_gradient(q));
phi.submit_gradient(gradient_coefficient, q);
}
phi.integrate_scatter(false, true, dst);
}
}
and
template <int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE
Tensor<1, n_components_to_use, Tensor<1, dim, Number>>
calculate_gradient_coefficient(
#if defined(USE_NONLINEAR) || defined(USE_ADVECTION)
const Tensor<1, n_components_to_use, Number> &input_value,
#endif
const Tensor<1, n_components_to_use, Tensor<1, dim, Number>> &
input_gradient){
Tensor<1, n_components_to_use, Tensor<1, dim, Number>> ret_val;
for(size_t component = 0; component < n_components_to_use; ++
component){
for(size_t d = 0; d < dim; ++d){
ret_val[component][d] = -1. / (2 * M_PI * M_PI) *
input_gradient[component][d];
}
}
return ret_val;
}
This approach works, and delivers correct results. Now I wanted to test the
same approach for the non-linear diffusion equation with f = -exp(-2 * pi^2
* t) * 0.5 * pi^2 * (-cos(2 * pi * (x - y)) - cos(2 * pi * (x + y)) + cos(2
* pi * x) + cos(2 * pi * y)), which should be the solution to grad(u (grad
u)) with u = exp(-pi^2*t) * sin(pi * x) * sin(pi * y). Thus, I changed the
routines to
template <int dim, int degree, int n_points_1d>
void LaplaceOperator<dim, degree, n_points_1d>::local_apply_cell(
const MatrixFree<dim, Number> & data,
vector_t & dst,
const vector_t &src,
const std::pair<unsigned int, unsigned int> & cell_range)
const
{
FEEvaluation<dim, degree, n_points_1d, n_components_to_use, Number>
phi(data);
for (unsigned int cell = cell_range.first; cell < cell_range.second;
++cell)
{
phi.reinit(cell);
phi.read_dof_values_plain(src);
phi.evaluate(true, true);
for(size_t q = 0; q < phi.n_q_points; ++q){
auto value = phi.get_value(q);
auto gradient = phi.get_gradient(q);
phi.submit_value(calculate_value_coefficient(value,
phi.
quadrature_point(q),
local_time), q
);
phi.submit_gradient(calculate_gradient_coefficient(value,
gradient
), q);
}
phi.integrate_scatter(true, true, dst);
}
}
and
template <int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE
Tensor<1, n_components_to_use, Number> calculate_value_coefficient(const
Tensor<1, n_components_to_use, Number> &input_value,
const
Point<dim, Number> &point,
const
double &time){
Tensor<1, n_components_to_use, Number> ret_val;
//(void) input_value;
(void) input_value;
for(size_t component = 0; component < n_components_to_use; ++
component){
const double x = point[component][0];
const double y = point[component][1];
ret_val[component] = (- exp(-2 * M_PI * M_PI * time)
* 0.5 * M_PI * M_PI * (-cos(2 * M_PI * (x
- y))
- cos(2 * M_PI * (x
+ y))
+ cos(2 * M_PI * x)
+ cos(2 * M_PI * y
)))
;
}
return ret_val;
}
template <int dim, typename Number>
inline DEAL_II_ALWAYS_INLINE
Tensor<1, n_components_to_use, Tensor<1, dim, Number>>
calculate_gradient_coefficient(
#if defined(USE_NONLINEAR) || defined(USE_ADVECTION)
const Tensor<1, n_components_to_use, Number> &input_value,
#endif
const Tensor<1, n_components_to_use, Tensor<1, dim, Number>> &
input_gradient){
Tensor<1, n_components_to_use, Tensor<1, dim, Number>> ret_val;
for(size_t component = 0; component < n_components_to_use; ++
component){
for(size_t d = 0; d < dim; ++d){
ret_val[component][d] = -1. * input_value[component] *
input_gradient[component][d];
}
}
return ret_val;
}
Unfortunately, now the result is not correct anymore. The initial sin-shape
is spreading out in both x- and y-direction, leading to wrong
results.Therefore I was wondering if I just implemented the functions in a
wrong way, or if there could be something else wrong?
Thanks!
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