num::pde Namespace Reference

Functions
void	diffusion_step_2d (Vector &u, int N, double coeff, Backend b=best_backend)
void	diffusion_step_2d_dirichlet (Vector &u, int N, double coeff, Backend b=best_backend)
void	diffusion_step_2d_4th_dirichlet (Vector &u, int N, double coeff, Backend b=best_backend)
void	diffusion_step_2d_dirichlet (ScalarField2D &g, double coeff, Backend b=best_backend)
void	diffusion_step_2d_4th_dirichlet (ScalarField2D &g, double coeff, Backend b=best_backend)
SparseMatrix	laplacian_sparse_2d (int N)
SparseMatrix	backward_euler_matrix (int N, double coeff)
SparseMatrix	backward_euler_matrix (const Grid2D &grid, double coeff)
	Grid2D overload – N is read from the grid.
LinearSolver	make_cg_solver (const SparseMatrix &A, real tol=1e-6)
Matrix	poisson2d_fd (const Matrix &f, int N)
	Solve -Delta u = f on (0,1)^2 (Dirichlet) via DST with FD eigenvalues.
Matrix	poisson2d (const Matrix &f, int N)
	Solve -Delta u = f on (0,1)^2 (Dirichlet) via DST with exact eigenvalues.

Function Documentation

backward_euler_matrix() [1/2]

SparseMatrix num::pde::backward_euler_matrix ( const Grid2D &  grid, double  coeff  )

inline

Grid2D overload – N is read from the grid.

Definition at line 109 of file diffusion.hpp.

References backward_euler_matrix(), and num::Grid2D::N.

backward_euler_matrix() [2/2]

SparseMatrix num::pde::backward_euler_matrix ( int  N, double  coeff  )

inline

Build the backward Euler system matrix A = I - coeff*L. Solve A * u^{n+1} = u^n at each time step.

Definition at line 90 of file diffusion.hpp.

References num::SparseMatrix::from_triplets().

Referenced by backward_euler_matrix().

diffusion_step_2d()

void num::pde::diffusion_step_2d ( Vector &  u, int  N, double  coeff, Backend  b = best_backend  )

inline

Explicit Euler diffusion step on a periodic 2D grid: u += coeff * Lap_periodic(u)

Definition at line 31 of file diffusion.hpp.

References num::axpy(), num::laplacian_stencil_2d_periodic(), and num::BasicVector< T >::size().

diffusion_step_2d_4th_dirichlet() [1/2]

void num::pde::diffusion_step_2d_4th_dirichlet ( ScalarField2D &  g, double  coeff, Backend  b = best_backend  )

inline

Definition at line 63 of file diffusion.hpp.

References diffusion_step_2d_4th_dirichlet(), num::ScalarField2D::N(), and num::ScalarField2D::vec().

diffusion_step_2d_4th_dirichlet() [2/2]

void num::pde::diffusion_step_2d_4th_dirichlet ( Vector &  u, int  N, double  coeff, Backend  b = best_backend  )

inline

Explicit Euler diffusion step with Dirichlet BCs and 4th-order Laplacian. Stability requires coeff <= ~0.08 (vs 0.25 for 2nd order).

Definition at line 49 of file diffusion.hpp.

References num::axpy(), num::laplacian_stencil_2d_4th(), and num::BasicVector< T >::size().

Referenced by diffusion_step_2d_4th_dirichlet().

diffusion_step_2d_dirichlet() [1/2]

void num::pde::diffusion_step_2d_dirichlet ( ScalarField2D &  g, double  coeff, Backend  b = best_backend  )

inline

Definition at line 58 of file diffusion.hpp.

References diffusion_step_2d_dirichlet(), num::ScalarField2D::N(), and num::ScalarField2D::vec().

diffusion_step_2d_dirichlet() [2/2]

void num::pde::diffusion_step_2d_dirichlet ( Vector &  u, int  N, double  coeff, Backend  b = best_backend  )

inline

Explicit Euler diffusion step with Dirichlet (zero) BCs: u += coeff * Lap_dirichlet(u)

Definition at line 40 of file diffusion.hpp.

References num::axpy(), num::laplacian_stencil_2d(), and num::BasicVector< T >::size().

Referenced by diffusion_step_2d_dirichlet().

laplacian_sparse_2d()

SparseMatrix num::pde::laplacian_sparse_2d ( int  N )

inline

Build the N^2 x N^2 sparse 5-point Laplacian matrix (Dirichlet BCs). Entry (k,k) = -4, off-diagonals = +1 for each interior neighbor.

Definition at line 70 of file diffusion.hpp.

References num::SparseMatrix::from_triplets().

make_cg_solver()

LinearSolver num::pde::make_cg_solver ( const SparseMatrix &  A, real  tol = 1e-6  )

inline

Returns a LinearSolver that solves A*x = rhs using matrix-free CG. Captures A by reference – A must outlive the returned callable.

Definition at line 115 of file diffusion.hpp.

References num::cg_matfree(), make_cg_solver(), num::matvec(), and num::sparse_matvec().

Referenced by make_cg_solver().

poisson2d()

Matrix num::pde::poisson2d	(	const Matrix &	f,
		int	N
	)

Solve -Delta u = f on (0,1)^2 (Dirichlet) via DST with exact eigenvalues.

Replaces the FD eigenvalue D_k = 2(1 - cos(k*pi/(N+1))) with the exact eigenvalue (k*pi)^2 of the continuous operator -d^2/dx^2. The error is determined by the quadrature approximation of the DST projection rather than the FD truncation of the eigenvalue, giving spectral accuracy.

Parameters

f	N x N matrix of RHS values.
N	Grid dimension. Must satisfy N = 2^p - 1.

Returns

N x N solution matrix u.

Definition at line 145 of file poisson.cpp.

References num::pi.

poisson2d_fd()

Matrix num::pde::poisson2d_fd	(	const Matrix &	f,
		int	N
	)

Solve -Delta u = f on (0,1)^2 (Dirichlet) via DST with FD eigenvalues.

Parameters

f	N x N matrix of RHS values on the interior grid (row i, col j corresponds to the grid point ((i+1)*h, (j+1)*h)).
N	Grid dimension. Must satisfy N = 2^p - 1.

Returns

N x N solution matrix u.

Definition at line 106 of file poisson.cpp.

References num::pi.