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advection_solver_irk.cc
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// This file is part of the advection_miniapp repository and subject to the
// LGPL license. See the LICENSE file in the top level directory of this
// project for details.
// Program for explicit time integration of the advection problem
// Author: Martin Kronbichler, Technical University of Munich, 2014-2020
//
// This program shares many similarities with the step-67 tutorial program of
// deal.II, see https://dealii.org/developer/doxygen/deal.II/step_67.html ,
// but it implements a simpler equation and is therefore ideal for learning
// about matrix-free evaluators.
// Compared to the main program advection_solver.cc, this program implements
// a variant with variable transport speed derived from an analytical
// expression. This file intentionally duplicates most of the other
// program to keep all implementation in a single file; for sustainable
// software design it would be advisable to share common code between the two
// cases.
#include <deal.II/base/function.h>
#include <deal.II/base/mpi.h>
#include <deal.II/base/timer.h>
#include <deal.II/base/utilities.h>
#include <deal.II/distributed/tria.h>
#include <deal.II/dofs/dof_handler.h>
#include <deal.II/dofs/dof_tools.h>
#include <deal.II/fe/fe_dgq.h>
#include <deal.II/fe/mapping_q_generic.h>
#include <deal.II/grid/grid_generator.h>
#include <deal.II/grid/grid_out.h>
#include <deal.II/grid/grid_tools.h>
#include <deal.II/grid/manifold_lib.h>
#include <deal.II/grid/tria.h>
#include <deal.II/lac/affine_constraints.h>
#include <deal.II/lac/la_parallel_block_vector.h>
#include <deal.II/lac/la_parallel_vector.h>
#include <deal.II/lac/lapack_full_matrix.h>
#include <deal.II/lac/precondition.h>
#include <deal.II/lac/solver_bicgstab.h>
#include <deal.II/lac/solver_gmres.h>
#include <deal.II/lac/solver_idr.h>
#include <deal.II/matrix_free/fe_evaluation.h>
#include <deal.II/matrix_free/matrix_free.h>
#include <deal.II/matrix_free/operators.h>
#include <deal.II/numerics/data_out.h>
#include <deal.II/numerics/point_value_history.h>
#include <deal.II/numerics/solution_transfer.h>
#include <deal.II/numerics/vector_tools.h>
#include <fstream>
#include <iomanip>
#include <iostream>
namespace DGAdvection
{
using namespace dealii;
// The dimension can be selected to be 1, 2, 3 (it is a C++ template
// argument, so different code gets compiled in 1D/2D/3D); for the advection
// speed of a rotating vortex that is set here, only dimension 2 is allowed
const unsigned int dimension = 2;
// The polynomial degree can be selected between 0 and any reasonable number
// (around 30), depending on the dimension and the mesh size
const unsigned int fe_degree = 6;
// The time step size is controlled via this parameter as
// dt = courant_number * min_h / (transport_norm * fe_degree^1.5)
const double courant_number = 2;
// Number of stages in IRK method
const unsigned int n_stages = 3;
// 0: central flux, 1: classical upwind flux (= Lax-Friedrichs)
const double flux_alpha = 1.0;
// The final simulation time
const double FINAL_TIME = 8;
// Frequency of output
const double output_tick = 0.1;
// Whether to mesh the domain with Cartesian mesh elements or with curved
// elements (more memory transfer -> slower)
enum class MeshType
{
cartesian,
deformed_cartesian,
inscribed_circle
};
constexpr MeshType mesh_type = MeshType::cartesian;
// Whether to set periodic boundary conditions on the domain (needs periodic
// solution as well)
const bool periodic = true;
// Switch to change between a conservative formulation of the advection term
// (factor 0) or a skew-symmetric one (factor 0.5)
const double factor_skew = 0.0;
// Switch to enable Gauss-Lobatto quadrature (true) or Gauss quadrature
// (false)
const bool use_gl_quad = false;
// Switch to enable Gauss--Lobatto quadrature for the inverse mass
// matrix. If false, use Gauss quadrature
const bool use_gl_quad_mass = false;
// Enable or disable writing of result files for visualization with ParaView
// or VisIt
const bool print_vtu = true;
// Analytical solution of the problem
template <int dim>
class ExactSolution : public Function<dim>
{
public:
ExactSolution(const double time = 0.)
: Function<dim>(1, time)
{}
virtual double
value(const Point<dim> &p,
const unsigned int /*component*/ = 0) const override
{
return value<double>(p);
}
template <typename Number>
Number
value(const Point<dim, Number> &p) const
{
return std::exp(-400. * ((p[0] - 0.5) * (p[0] - 0.5) +
(p[1] - 0.75) * (p[1] - 0.75))) +
std::exp(-100. * ((p[0] - 0.5) * (p[0] - 0.5) +
(p[1] - 0.25) * (p[1] - 0.25)));
}
};
template <int dim>
class TransportSpeed
{
public:
TransportSpeed(const double time)
: time(time)
{}
template <typename Number>
Tensor<1, dim, Number>
value(const Point<dim, Number> &p) const
{
const double factor = std::cos(numbers::PI * time / FINAL_TIME) * 2.;
Tensor<1, dim, Number> result;
result[0] = factor * (std::sin(2 * numbers::PI * p[1]) *
std::sin(numbers::PI * p[0]) *
std::sin(numbers::PI * p[0]) +
0.2 * std::sin(20 * numbers::PI * (p[0] + 0.2)) *
std::cos(20 * numbers::PI * (p[1] + 0.3)));
result[1] = -factor * (std::sin(2 * numbers::PI * p[0]) *
std::sin(numbers::PI * p[1]) *
std::sin(numbers::PI * p[1]) +
0.2 * std::cos(20 * numbers::PI * (p[0] + 0.2)) *
std::sin(20 * numbers::PI * (p[1] + 0.3)));
return result;
}
private:
const double time;
};
// Description of curved mesh
template <int dim>
class DeformedCubeManifold : public ChartManifold<dim, dim, dim>
{
public:
DeformedCubeManifold(const double left,
const double right,
const double deformation,
const unsigned int frequency = 1)
: left(left)
, right(right)
, deformation(deformation)
, frequency(frequency)
{}
Point<dim>
push_forward(const Point<dim> &chart_point) const override
{
double sinval = deformation;
for (unsigned int d = 0; d < dim; ++d)
sinval *= std::sin(frequency * numbers::PI * (chart_point(d) - left) /
(right - left));
Point<dim> space_point;
for (unsigned int d = 0; d < dim; ++d)
space_point(d) = chart_point(d) + sinval;
return space_point;
}
Point<dim>
pull_back(const Point<dim> &space_point) const override
{
Point<dim> x = space_point;
Point<dim> one;
for (unsigned int d = 0; d < dim; ++d)
one(d) = 1.;
// Newton iteration to solve the nonlinear equation given by the point
Tensor<1, dim> sinvals;
for (unsigned int d = 0; d < dim; ++d)
sinvals[d] =
std::sin(frequency * numbers::PI * (x(d) - left) / (right - left));
double sinval = deformation;
for (unsigned int d = 0; d < dim; ++d)
sinval *= sinvals[d];
Tensor<1, dim> residual = space_point - x - sinval * one;
unsigned int its = 0;
while (residual.norm() > 1e-12 && its < 100)
{
Tensor<2, dim> jacobian;
for (unsigned int d = 0; d < dim; ++d)
jacobian[d][d] = 1.;
for (unsigned int d = 0; d < dim; ++d)
{
double sinval_der = deformation * frequency / (right - left) *
numbers::PI *
std::cos(frequency * numbers::PI *
(x(d) - left) / (right - left));
for (unsigned int e = 0; e < dim; ++e)
if (e != d)
sinval_der *= sinvals[e];
for (unsigned int e = 0; e < dim; ++e)
jacobian[e][d] += sinval_der;
}
x += invert(jacobian) * residual;
for (unsigned int d = 0; d < dim; ++d)
sinvals[d] = std::sin(frequency * numbers::PI * (x(d) - left) /
(right - left));
sinval = deformation;
for (unsigned int d = 0; d < dim; ++d)
sinval *= sinvals[d];
residual = space_point - x - sinval * one;
++its;
}
AssertThrow(residual.norm() < 1e-12,
ExcMessage("Newton for point did not converge."));
return x;
}
std::unique_ptr<Manifold<dim>>
clone() const override
{
return std::make_unique<DeformedCubeManifold<dim>>(left,
right,
deformation,
frequency);
}
private:
const double left;
const double right;
const double deformation;
const unsigned int frequency;
};
std::shared_ptr<const Utilities::MPI::Partitioner>
create_partitioner_multiple(
const std::shared_ptr<const Utilities::MPI::Partitioner>
&scalar_partitioner,
const unsigned int multiplicity)
{
IndexSet owned(multiplicity * scalar_partitioner->size());
owned.add_range(multiplicity * scalar_partitioner->local_range().first,
multiplicity * scalar_partitioner->local_range().second);
IndexSet ghosted(owned.size());
for (auto it = scalar_partitioner->ghost_indices().begin_intervals();
it != scalar_partitioner->ghost_indices().end_intervals();
++it)
ghosted.add_range(multiplicity * (*it->begin()),
multiplicity * (it->last() + 1));
return std::make_shared<Utilities::MPI::Partitioner>(
owned, ghosted, scalar_partitioner->get_mpi_communicator());
}
class IRK
{
public:
IRK()
{
A.reinit(n_stages, n_stages);
if (n_stages == 1)
A(0, 0) = 1.;
else if (n_stages == 2)
{
A(0, 0) = 5. / 12.;
A(0, 1) = -1. / 12.;
A(1, 0) = 3. / 4.;
A(1, 1) = 1. / 4.;
}
else if (n_stages == 3)
{
A(0, 0) = 0.19681547722366044;
A(0, 1) = -0.06553542585019838;
A(0, 2) = 0.02377097434822015;
A(1, 0) = 0.3944243147390873;
A(1, 1) = 0.29207341166522843;
A(1, 2) = -0.04154875212599792;
A(2, 0) = 0.37640306270046725;
A(2, 1) = 0.5124858261884216;
A(2, 2) = 1. / 9.;
}
else if (n_stages == 4)
{
A(0, 0) = 0.11299954065227377;
A(0, 1) = -0.04030922142462728;
A(0, 2) = 0.025802391441157962;
A(0, 3) = -0.009904667574594527;
A(1, 0) = 0.2343839541547278;
A(1, 1) = 0.20689257722452742;
A(1, 2) = -0.047857142857142855;
A(1, 3) = 0.016047419401474628;
A(2, 0) = 0.21668177697189483;
A(2, 1) = 0.4061232017705644;
A(2, 2) = 0.18903654485049834;
A(2, 3) = -0.02418210484824596;
A(3, 0) = 0.22046204620462045;
A(3, 1) = 0.3881932021466905;
A(3, 2) = 0.3288439955106622;
A(3, 3) = 1. / 16.;
}
else if (n_stages == 5)
{
A(0, 0) = 0.072998864317903;
A(0, 1) = -0.026735331107945;
A(0, 2) = 0.018676929763984;
A(0, 3) = -0.012879106093306;
A(0, 4) = 0.005042839233882;
A(1, 0) = 0.153775231479182;
A(1, 1) = 0.146214867847493;
A(1, 2) = -0.036444568905128;
A(1, 3) = 0.021233063119304;
A(1, 4) = -0.007935579902729;
A(2, 0) = 0.140063045684809;
A(2, 1) = 0.298967129491283;
A(2, 2) = 0.167585070135249;
A(2, 3) = -0.033969101686618;
A(2, 4) = 0.010944288744193;
A(3, 0) = 0.144894308109531;
A(3, 1) = 0.276500068760161;
A(3, 2) = 0.325797922910423;
A(3, 3) = 0.128756753254908;
A(3, 4) = -0.015708917378805;
A(4, 0) = 0.143713560791217;
A(4, 1) = 0.281356015149465;
A(4, 2) = 0.311826522975736;
A(4, 3) = 0.223103901083572;
A(4, 4) = 1. / 25.;
}
b.reinit(A.m());
for (unsigned int i = 0; i < b.size(); ++i)
b(i) = A(A.m() - 1, i);
c.reinit(A.m());
for (unsigned int i = 0; i < c.size(); ++i)
for (unsigned int j = 0; j < c.size(); ++j)
c(i) += A(i, j);
inv_A = A;
inv_A.gauss_jordan();
LAPACKFullMatrix<double> A_lap;
A_lap.copy_from(inv_A);
A_lap.compute_eigenvalues(true, false);
eigenvalues.resize(A.n());
for (unsigned int i = 0; i < A.n(); ++i)
eigenvalues[i] = A_lap.eigenvalue(i);
T = A_lap.get_right_eigenvectors();
inv_T = T;
inv_T.gauss_jordan();
}
FullMatrix<double> A, inv_A;
Vector<double> b;
Vector<double> c;
std::vector<std::complex<double>> eigenvalues;
FullMatrix<std::complex<double>> T, inv_T;
};
template <int dim, int fe_degree, int n_stages, typename Number = double>
class CellwiseOperator
{
public:
CellwiseOperator(
const Tensor<2, dim, VectorizedArray<Number>> &jac,
const internal::MatrixFreeFunctions::UnivariateShapeData<
VectorizedArray<Number>> &shape,
const Tensor<1, dim, VectorizedArray<Number>> *speed_cells,
const Table<2, VectorizedArray<Number>> &normal_speed_faces,
const Quadrature<dim> &cell_quadrature,
const Quadrature<dim - 1> &face_quadrature,
const double inv_dt,
const FullMatrix<double> &inv_A,
const std::array<Number, n_stages> &time_factor)
: jac(jac)
, shape(shape)
, speed_cells(speed_cells)
, normal_speed_faces(normal_speed_faces)
, cell_quadrature(cell_quadrature)
, face_quadrature(face_quadrature)
, inv_dt(inv_dt)
, inv_A(inv_A)
, time_factors(time_factor)
{}
void
vmult(Vector<Number> &dst, const Vector<Number> &src) const
{
const VectorizedArray<Number> *src_ptr =
reinterpret_cast<const VectorizedArray<Number> *>(src.data());
VectorizedArray<Number> *dst_ptr =
reinterpret_cast<VectorizedArray<Number> *>(dst.data());
const unsigned int dofs_per_component =
Utilities::pow(fe_degree + 1, dim);
VectorizedArray<Number> gradients[n_stages][dim][dofs_per_component];
// face integrals relevant to present cell
for (unsigned int d = 0; d < dim; ++d)
{
const unsigned int stride = Utilities::pow(fe_degree + 1, d);
const unsigned int offset0 = d > 0 ? 1 : fe_degree + 1;
const unsigned int offset1 =
(dim > 2 ?
(d == 2 ? (fe_degree + 1) : Utilities::pow(fe_degree + 1, 2)) :
1);
VectorizedArray<Number> surface_JxW = 1.;
for (unsigned int e = 0; e < dim; ++e)
if (d != e)
surface_JxW *= jac[e][e];
surface_JxW = 1.0 / surface_JxW;
for (unsigned int i1 = 0; i1 < (dim > 2 ? fe_degree + 1 : 1); ++i1)
for (unsigned int i0 = 0; i0 < (dim > 1 ? fe_degree + 1 : 1); ++i0)
{
VectorizedArray<Number> sum_l[n_stages] = {},
sum_r[n_stages] = {};
const auto *my_ptr = src_ptr + offset1 * i1 + offset0 * i0;
for (unsigned int j = 0; j < fe_degree + 1;
++j, my_ptr += stride)
{
for (unsigned int stage = 0; stage < n_stages; ++stage)
{
const VectorizedArray<Number> value =
my_ptr[stage * dofs_per_component];
sum_l[stage] +=
shape.quadrature_data_on_face[0][j] * value;
sum_r[stage] +=
shape.quadrature_data_on_face[0][fe_degree - j] *
value;
}
}
const unsigned int q = (dim == 3 && d == 1) ?
i0 * (fe_degree + 1) + i1 :
i1 * (fe_degree + 1) + i0;
{
for (unsigned int stage = 0; stage < n_stages; ++stage)
{
const auto speed =
normal_speed_faces(2 * d, q) * time_factors[stage];
const auto coefficient =
0.5 * (speed + flux_alpha * std::abs(speed)) -
factor_skew * speed;
sum_l[stage] = sum_l[stage] * coefficient *
(face_quadrature.weight(q) * surface_JxW);
}
}
{
for (unsigned int stage = 0; stage < n_stages; ++stage)
{
const auto speed =
normal_speed_faces(2 * d + 1, q) * time_factors[stage];
const auto coefficient =
0.5 * (speed + flux_alpha * std::abs(speed)) -
factor_skew * speed;
sum_r[stage] = sum_r[stage] * coefficient *
(face_quadrature.weight(q) * surface_JxW);
}
}
auto out_ptr = dst_ptr + offset1 * i1 + offset0 * i0;
if (d == 0)
for (unsigned int j = 0; j < fe_degree + 1;
++j, out_ptr += stride)
for (unsigned int stage = 0; stage < n_stages; ++stage)
out_ptr[stage * dofs_per_component] =
shape.quadrature_data_on_face[0][j] * sum_l[stage] +
shape.quadrature_data_on_face[0][fe_degree - j] *
sum_r[stage];
else
for (unsigned int j = 0; j < fe_degree + 1;
++j, out_ptr += stride)
for (unsigned int stage = 0; stage < n_stages; ++stage)
out_ptr[stage * dofs_per_component] +=
shape.quadrature_data_on_face[0][j] * sum_l[stage] +
shape.quadrature_data_on_face[0][fe_degree - j] *
sum_r[stage];
}
}
internal::EvaluatorTensorProduct<internal::evaluate_evenodd,
dim,
fe_degree + 1,
fe_degree + 1,
VectorizedArray<Number>>
eval(shape.shape_values_eo,
shape.shape_gradients_collocation_eo,
AlignedVector<VectorizedArray<Number>>());
// volume integrals
for (unsigned int stage = 0; stage < n_stages; ++stage)
{
eval.template gradients<0, true, false>(src_ptr +
stage * dofs_per_component,
gradients[stage][0]);
if (dim > 1)
eval.template gradients<1, true, false>(
src_ptr + stage * dofs_per_component, gradients[stage][1]);
if (dim > 2)
eval.template gradients<2, true, false>(
src_ptr + stage * dofs_per_component, gradients[stage][2]);
}
const VectorizedArray<Number> JxW = 1. / determinant(jac);
for (unsigned int q = 0; q < dofs_per_component; ++q)
{
const auto speed = speed_cells[q];
VectorizedArray<Number> u[n_stages];
VectorizedArray<Number> flux[n_stages];
for (unsigned int stage = 0; stage < n_stages; ++stage)
{
u[stage] = src_ptr[q + stage * dofs_per_component];
VectorizedArray<Number> speed_gradu =
speed[0] * gradients[stage][0][q] * jac[0][0];
for (unsigned int d = 1; d < dim; ++d)
speed_gradu += speed[d] * gradients[stage][d][q] * jac[d][d];
flux[stage] = (factor_skew * time_factors[stage]) * speed_gradu;
const VectorizedArray<Number> result =
(-1.0 + factor_skew) * time_factors[stage] * u[stage] *
(JxW * cell_quadrature.weight(q));
for (unsigned int d = 0; d < dim; ++d)
gradients[stage][d][q] = result * speed[d] * jac[d][d];
}
// mass matrix part
for (unsigned int stage = 0; stage < n_stages; ++stage)
u[stage] *= VectorizedArray<Number>(inv_dt);
for (unsigned int stage = 0; stage < n_stages; ++stage)
{
for (unsigned int s = 0; s < n_stages; ++s)
flux[stage] += inv_A(stage, s) * u[s];
dst_ptr[q + stage * dofs_per_component] +=
flux[stage] * (JxW * cell_quadrature.weight(q));
}
}
for (unsigned int stage = 0; stage < n_stages; ++stage)
{
eval.template gradients<0, false, true>(gradients[stage][0],
dst_ptr +
stage * dofs_per_component);
if (dim > 1)
eval.template gradients<1, false, true>(
gradients[stage][1], dst_ptr + stage * dofs_per_component);
if (dim > 2)
eval.template gradients<2, false, true>(
gradients[stage][2], dst_ptr + stage * dofs_per_component);
}
}
void
transform_to_collocation(const VectorizedArray<Number> *src_ptr,
Vector<Number> &dst) const
{
internal::EvaluatorTensorProduct<internal::evaluate_evenodd,
dim,
fe_degree + 1,
fe_degree + 1,
VectorizedArray<Number>>
evaluator(AlignedVector<VectorizedArray<Number>>(),
AlignedVector<VectorizedArray<Number>>(),
shape.inverse_shape_values_eo);
VectorizedArray<Number> *dst_ptr =
reinterpret_cast<VectorizedArray<Number> *>(dst.data());
const unsigned int dofs_per_component =
Utilities::pow(fe_degree + 1, dim);
for (unsigned int d = 0; d < n_stages; ++d)
{
const VectorizedArray<Number> *in = src_ptr + d * dofs_per_component;
VectorizedArray<Number> *out = dst_ptr + d * dofs_per_component;
// Need to select 'apply' method with hessian slot because values
// assume symmetries that do not exist in the inverse shapes
evaluator.template hessians<0, true, false>(in, out);
if (dim > 1)
evaluator.template hessians<1, true, false>(out, out);
if (dim > 2)
evaluator.template hessians<2, true, false>(out, out);
}
}
void
transform_from_collocation(const Vector<Number> &src,
VectorizedArray<Number> *dst_ptr) const
{
internal::EvaluatorTensorProduct<internal::evaluate_evenodd,
dim,
fe_degree + 1,
fe_degree + 1,
VectorizedArray<Number>>
evaluator(AlignedVector<VectorizedArray<Number>>(),
AlignedVector<VectorizedArray<Number>>(),
shape.inverse_shape_values_eo);
const VectorizedArray<Number> *src_ptr =
reinterpret_cast<const VectorizedArray<Number> *>(src.data());
const unsigned int dofs_per_component =
Utilities::pow(fe_degree + 1, dim);
for (unsigned int d = 0; d < n_stages; ++d)
{
const VectorizedArray<Number> *in = src_ptr + d * dofs_per_component;
VectorizedArray<Number> *out = dst_ptr + d * dofs_per_component;
// Need to select 'apply' method with hessian slot because values
// assume symmetries that do not exist in the inverse shapes
evaluator.template hessians<0, false, false>(in, out);
if (dim > 1)
evaluator.template hessians<1, false, false>(out, out);
if (dim > 2)
evaluator.template hessians<2, false, false>(out, out);
}
}
private:
const Tensor<2, dim, VectorizedArray<Number>> jac;
const internal::MatrixFreeFunctions::UnivariateShapeData<
VectorizedArray<Number>> &shape;
const Tensor<1, dim, VectorizedArray<Number>> *speed_cells;
const Table<2, VectorizedArray<Number>> &normal_speed_faces;
const Quadrature<dim> &cell_quadrature;
const Quadrature<dim - 1> &face_quadrature;
const Number inv_dt;
const FullMatrix<double> &inv_A;
const std::array<Number, n_stages> time_factors;
};
template <int dim, int fe_degree, int n_stages, typename Number = double>
class CellwiseRungeKutta
{
public:
CellwiseRungeKutta(
const Tensor<2, dim, VectorizedArray<Number>> &jac,
const internal::MatrixFreeFunctions::UnivariateShapeData<
VectorizedArray<Number>> &shape,
const Tensor<1, dim, VectorizedArray<Number>> *speed_cells,
const Table<2, VectorizedArray<Number>> &normal_speed_faces,
const Quadrature<dim> &cell_quadrature,
const Quadrature<dim - 1> &face_quadrature,
const double dt,
const FullMatrix<double> &A,
const std::array<Number, n_stages> &time_factor)
: jac(jac)
, shape(shape)
, speed_cells(speed_cells)
, normal_speed_faces(normal_speed_faces)
, cell_quadrature(cell_quadrature)
, face_quadrature(face_quadrature)
, dt(dt)
, A(A)
, time_factors(time_factor)
{}
void
propagate(VectorizedArray<Number> *values) const
{
std::array<Number, 3> bi{
{0.245170287303492, 0.184896052186740, 0.569933660509768}};
std::array<Number, 2> ai{{0.755726351946097, 0.386954477304099}};
const unsigned int n_dofs = Utilities::pow(fe_degree + 1, dim) * n_stages;
VectorizedArray<Number> sol_tmp[n_dofs];
VectorizedArray<Number> ki[n_dofs];
for (unsigned int i = 0; i < n_dofs; ++i)
values[i] = values[i] * dt;
apply_inverse_mass(values);
VectorizedArray<Number> max_dt = 0.;
for (unsigned int q = 0; q < cell_quadrature.size(); ++q)
max_dt = std::max((jac * speed_cells[q]).norm_square(), max_dt);
max_dt *= std::max(std::abs(time_factors[0]),
std::abs(time_factors[n_stages - 1]));
for (unsigned int v = 1; v < VectorizedArray<Number>::size(); ++v)
max_dt[0] = std::max(max_dt[0], max_dt[v]);
Number my_dt = std::min(
dt, 0.05 / std::sqrt(Utilities::pow(fe_degree + 1, 3) * max_dt[0]));
const unsigned int n_steps =
static_cast<unsigned int>(std::round(dt / my_dt));
my_dt = dt / n_steps;
// std::cout << my_dt << " " << n_steps << " ";
for (unsigned int step = 0; step < n_steps; ++step)
{
apply(values, ki);
for (unsigned int i = 0; i < n_dofs; ++i)
{
sol_tmp[i] = values[i] + (my_dt * ai[0]) * ki[i];
values[i] += (my_dt * bi[0]) * ki[i];
}
apply(sol_tmp, ki);
for (unsigned int i = 0; i < n_dofs; ++i)
{
sol_tmp[i] = values[i] + (my_dt * ai[1]) * ki[i];
values[i] += (my_dt * bi[1]) * ki[i];
}
apply(sol_tmp, ki);
for (unsigned int i = 0; i < n_dofs; ++i)
values[i] += (my_dt * bi[2]) * ki[i];
}
}
void
apply(const VectorizedArray<Number> *src_ptr,
VectorizedArray<Number> *dst_ptr) const
{
constexpr unsigned int dofs_per_component =
Utilities::pow(fe_degree + 1, dim);
VectorizedArray<Number> gradients[dim][dofs_per_component];
internal::EvaluatorTensorProduct<internal::evaluate_evenodd,
dim,
fe_degree + 1,
fe_degree + 1,
VectorizedArray<Number>>
eval(shape.shape_values_eo,
shape.shape_gradients_collocation_eo,
AlignedVector<VectorizedArray<Number>>());
// face integrals relevant to present cell
for (unsigned int d = 0; d < dim; ++d)
{
const unsigned int stride = Utilities::pow(fe_degree + 1, d);
const unsigned int offset0 = d > 0 ? 1 : fe_degree + 1;
const unsigned int offset1 =
(dim > 2 ?
(d == 2 ? (fe_degree + 1) : Utilities::pow(fe_degree + 1, 2)) :
1);
VectorizedArray<Number> surface_JxW = 1.;
for (unsigned int e = 0; e < dim; ++e)
if (d != e)
surface_JxW *= jac[e][e];
surface_JxW = 1.0 / surface_JxW;
for (unsigned int i1 = 0; i1 < (dim > 2 ? fe_degree + 1 : 1); ++i1)
for (unsigned int i0 = 0; i0 < (dim > 1 ? fe_degree + 1 : 1); ++i0)
{
VectorizedArray<Number> sum_l[n_stages] = {},
sum_r[n_stages] = {};
const auto *my_ptr = src_ptr + offset1 * i1 + offset0 * i0;
for (unsigned int j = 0; j < fe_degree + 1;
++j, my_ptr += stride)
{
for (unsigned int stage = 0; stage < n_stages; ++stage)
{
const VectorizedArray<Number> value =
my_ptr[stage * dofs_per_component];
sum_l[stage] +=
shape.quadrature_data_on_face[0][j] * value;
sum_r[stage] +=
shape.quadrature_data_on_face[0][fe_degree - j] *
value;
}
}
const unsigned int q = (dim == 3 && d == 1) ?
i0 * (fe_degree + 1) + i1 :
i1 * (fe_degree + 1) + i0;
{
for (unsigned int stage = 0; stage < n_stages; ++stage)
{
const auto speed =
normal_speed_faces(2 * d, q) * time_factors[stage];
const auto coefficient =
0.5 * (speed + flux_alpha * std::abs(speed)) -
factor_skew * speed;
sum_l[stage] = sum_l[stage] * coefficient *
(face_quadrature.weight(q) * surface_JxW);
}
}
{
for (unsigned int stage = 0; stage < n_stages; ++stage)
{
const auto speed =
normal_speed_faces(2 * d + 1, q) * time_factors[stage];
const auto coefficient =
0.5 * (speed + flux_alpha * std::abs(speed)) -
factor_skew * speed;
sum_r[stage] = sum_r[stage] * coefficient *
(face_quadrature.weight(q) * surface_JxW);
}
}
auto out_ptr = dst_ptr + offset1 * i1 + offset0 * i0;
if (d == 0)
for (unsigned int j = 0; j < fe_degree + 1;
++j, out_ptr += stride)
for (unsigned int stage = 0; stage < n_stages; ++stage)
out_ptr[stage * dofs_per_component] =
shape.quadrature_data_on_face[0][j] * sum_l[stage] +
shape.quadrature_data_on_face[0][fe_degree - j] *
sum_r[stage];
else
for (unsigned int j = 0; j < fe_degree + 1;
++j, out_ptr += stride)
for (unsigned int stage = 0; stage < n_stages; ++stage)
out_ptr[stage * dofs_per_component] +=
shape.quadrature_data_on_face[0][j] * sum_l[stage] +
shape.quadrature_data_on_face[0][fe_degree - j] *
sum_r[stage];
}
}
const VectorizedArray<Number> JxW = 1. / determinant(jac);
// volume integrals
for (unsigned int stage = 0; stage < n_stages; ++stage)
{
eval.template gradients<0, true, false>(src_ptr +
stage * dofs_per_component,
gradients[0]);
if (dim > 1)
eval.template gradients<1, true, false>(
src_ptr + stage * dofs_per_component, gradients[1]);
if (dim > 2)
eval.template gradients<2, true, false>(
src_ptr + stage * dofs_per_component, gradients[2]);
for (unsigned int q = 0; q < dofs_per_component; ++q)
{
const auto speed = speed_cells[q];
const auto u = src_ptr[q + stage * dofs_per_component];
VectorizedArray<Number> speed_gradu =
speed[0] * gradients[0][q] * jac[0][0];
for (unsigned int d = 1; d < dim; ++d)
speed_gradu += speed[d] * gradients[d][q] * jac[d][d];
dst_ptr[q + stage * dofs_per_component] +=
(factor_skew * time_factors[stage]) * speed_gradu *
(JxW * cell_quadrature.weight(q));
const VectorizedArray<Number> result =
(-1.0 + factor_skew) * time_factors[stage] * u *
(JxW * cell_quadrature.weight(q));
for (unsigned int d = 0; d < dim; ++d)
gradients[d][q] = result * speed[d] * jac[d][d];
}
eval.template gradients<0, false, true>(gradients[0],
dst_ptr +
stage * dofs_per_component);
if (dim > 1)
eval.template gradients<1, false, true>(
gradients[1], dst_ptr + stage * dofs_per_component);
if (dim > 2)
eval.template gradients<2, false, true>(
gradients[2], dst_ptr + stage * dofs_per_component);
}
for (unsigned int i = 0; i < n_stages * dofs_per_component; ++i)
dst_ptr[i] = -dst_ptr[i];
apply_inverse_mass(dst_ptr);
}
void
apply_inverse_mass(VectorizedArray<Number> *dst_ptr) const
{
constexpr unsigned int dofs_per_component =
Utilities::pow(fe_degree + 1, dim);
const VectorizedArray<Number> inverse_jacobian_determinant =
determinant(jac);
VectorizedArray<Number> myA[n_stages][n_stages];
for (unsigned int i = 0; i < n_stages; ++i)
for (unsigned int j = 0; j < n_stages; ++j)
myA[i][j] = A(i, j);
for (unsigned int q = 0; q < dofs_per_component; ++q)
{
const VectorizedArray<Number> factor =
(1. / cell_quadrature.weight(q)) * inverse_jacobian_determinant;
VectorizedArray<Number> rhs[n_stages];
for (unsigned int stage = 0; stage < n_stages; ++stage)
rhs[stage] = dst_ptr[q + stage * dofs_per_component];
for (unsigned int stage = 0; stage < n_stages; ++stage)
{
VectorizedArray<Number> sum = myA[stage][0] * rhs[0];
for (unsigned int s = 1; s < n_stages; ++s)
sum += myA[stage][s] * rhs[s];
dst_ptr[q + stage * dofs_per_component] = sum * factor;
}
}
}
void
transform_to_collocation(const VectorizedArray<Number> *src_ptr,
VectorizedArray<Number> *dst_ptr) const
{
internal::EvaluatorTensorProduct<internal::evaluate_evenodd,
dim,
fe_degree + 1,
fe_degree + 1,
VectorizedArray<Number>>
evaluator(AlignedVector<VectorizedArray<Number>>(),
AlignedVector<VectorizedArray<Number>>(),
shape.inverse_shape_values_eo);
const unsigned int dofs_per_component =
Utilities::pow(fe_degree + 1, dim);
for (unsigned int d = 0; d < n_stages; ++d)
{
const VectorizedArray<Number> *in = src_ptr + d * dofs_per_component;
VectorizedArray<Number> *out = dst_ptr + d * dofs_per_component;
// Need to select 'apply' method with hessian slot because values
// assume symmetries that do not exist in the inverse shapes
evaluator.template hessians<0, true, false>(in, out);
if (dim > 1)
evaluator.template hessians<1, true, false>(out, out);
if (dim > 2)
evaluator.template hessians<2, true, false>(out, out);
}
}
void
transform_from_collocation(const VectorizedArray<Number> *src_ptr,
VectorizedArray<Number> *dst_ptr) const
{
internal::EvaluatorTensorProduct<internal::evaluate_evenodd,
dim,
fe_degree + 1,
fe_degree + 1,
VectorizedArray<Number>>
evaluator(AlignedVector<VectorizedArray<Number>>(),
AlignedVector<VectorizedArray<Number>>(),
shape.inverse_shape_values_eo);
const unsigned int dofs_per_component =
Utilities::pow(fe_degree + 1, dim);
for (unsigned int d = 0; d < n_stages; ++d)
{
const VectorizedArray<Number> *in = src_ptr + d * dofs_per_component;
VectorizedArray<Number> *out = dst_ptr + d * dofs_per_component;
// Need to select 'apply' method with hessian slot because values
// assume symmetries that do not exist in the inverse shapes
evaluator.template hessians<0, false, false>(in, out);
if (dim > 1)
evaluator.template hessians<1, false, false>(out, out);
if (dim > 2)
evaluator.template hessians<2, false, false>(out, out);
}
}
private:
const Tensor<2, dim, VectorizedArray<Number>> jac;
const internal::MatrixFreeFunctions::UnivariateShapeData<
VectorizedArray<Number>> &shape;
const Tensor<1, dim, VectorizedArray<Number>> *speed_cells;
const Table<2, VectorizedArray<Number>> &normal_speed_faces;
const Quadrature<dim> &cell_quadrature;
const Quadrature<dim - 1> &face_quadrature;
const Number dt;
const FullMatrix<double> &A;
const std::array<Number, n_stages> time_factors;
};