time_integration – cfd-solver

SSPRK = Strong Stability Preserving Runge-Kutta (Shu & Osher, 1988).

Supported schemes: 'euler' — Explicit Euler (1st order) 'ssprk22' — SSPRK(2,2) (Shu & Osher, 1988) 'rk3' — TVD-RK3 (Shu & Osher, 1988) [default] 'rk4' — Classic RK4 'ssprk54' — SSPRK(5,4) (Spiteri & Ruuth, 2002) 'beuler' — Backward Euler with Newton-Raphson (implicit, 1st order) 'bdf2' — BDF2 with Newton-Raphson (implicit, 2nd order)

Implicit banded solver selection is controlled per-instance via state%cfg%lapack_solver = .true. (default) — LAPACK dgbsv (pivoted, faster) state%cfg%lapack_solver = .false. — built-in Gaussian elimination (no pivoting) Abstract interface satisfied by every single-step stepper.

Uses

Variables

Type	Visibility	Attributes		Name		Initial
procedure(stepper_iface),	public,	pointer	::	step	=>	null()	Procedure pointer to the active time-stepping scheme. Initialised to null; set once by init_time_scheme() before the time loop.
integer,	private,	parameter	::	n_newton	=	3	Maximum Newton-Raphson iterations per time step (beuler and bdf2). 3 iterations is sufficient for BDF2 at CFL ≤ 10; increase for stiff problems.
real(kind=wp),	private,	parameter	::	tol_newton	=	1.0e-10_wp	Newton convergence tolerance: max-norm of correction ΔQ must fall below this. 1e-10 gives full convergence well below solver tolerance for double precision.
real(kind=wp),	private,	parameter	::	eps_jac	=	1.0e-7_wp	Finite-difference step size for Jacobian column approximation. ≈ √(machine ε) ≈ √(2.2e-16) ≈ 1.5e-8; balances FD truncation error (O(h)) against floating-point cancellation error (O(ε/h)).

Abstract Interfaces

abstract interface

public subroutine stepper_iface(state)

Arguments

Type	Intent	Optional	Attributes		Name
type(solver_state_t),	intent(inout)			::	state

Subroutines

public subroutine resolve_time_scheme(stepper, scheme)

Resolve a scheme name to a specific stepper procedure pointer.

Arguments

Type	Intent	Optional	Attributes		Name
procedure(stepper_iface),	intent(out),		pointer	::	stepper
character(len=*),	intent(in)			::	scheme

public subroutine init_time_scheme(scheme)

Bind the procedure pointer @p step to the requested scheme.

Arguments

Type	Intent	Optional	Attributes		Name
character(len=*),	intent(in)			::	scheme

private subroutine tvd_rk3_step(state)

Advance the solution by one time step using the TVD-RK3 scheme.

Arguments

Type	Intent	Optional	Attributes		Name
type(solver_state_t),	intent(inout)			::	state

private subroutine ssprk54_step(state)

Advance the solution by one time step using the SSPRK(5,4) scheme.

Arguments

Type	Intent	Optional	Attributes		Name
type(solver_state_t),	intent(inout)			::	state

private subroutine euler_step(state)

Advance the solution by one time step using the explicit Euler scheme.

Arguments

Type	Intent	Optional	Attributes		Name
type(solver_state_t),	intent(inout)			::	state

private subroutine ssprk22_step(state)

Advance the solution by one time step using the SSPRK(2,2) scheme.

Arguments

Type	Intent	Optional	Attributes		Name
type(solver_state_t),	intent(inout)			::	state

private subroutine rk4_step(state)

Advance the solution by one time step using the classic RK4 scheme.

Arguments

Type	Intent	Optional	Attributes		Name
type(solver_state_t),	intent(inout)			::	state

private subroutine setup_band_storage(state, n_dof, kl, ku, ldab, diag_row)

Compute banded-storage dimensions for the Newton-step Jacobian.

Arguments

Type	Intent		Name
type(solver_state_t),	intent(in)	::	state	Solver state (reads n_pt and cfg%lapack_solver).
integer,	intent(out)	::	n_dof	Total degrees of freedom (neq * n_pt)
integer,	intent(out)	::	kl	Total degrees of freedom (neq * n_pt) Lower bandwidth
integer,	intent(out)	::	ku	Total degrees of freedom (neq * n_pt) Lower bandwidth Upper bandwidth
integer,	intent(out)	::	ldab	Total degrees of freedom (neq * n_pt) Lower bandwidth Upper bandwidth Leading dimension of band matrix
integer,	intent(out)	::	diag_row	Total degrees of freedom (neq * n_pt) Lower bandwidth Upper bandwidth Leading dimension of band matrix Band-storage row that holds the diagonal

private subroutine band_lapack_solve(ab, kl, ku, n, b, info)

Banded linear solver using LAPACK dgbsv.

Arguments

Type	Intent		Name
real(kind=wp),	intent(inout)	::	ab(2*kl+ku+1,n)
integer,	intent(in)	::	kl	Lower bandwidth
integer,	intent(in)	::	ku	Lower bandwidth Upper bandwidth
integer,	intent(in)	::	n	Lower bandwidth Upper bandwidth Matrix order (number of unknowns)
real(kind=wp),	intent(inout)	::	b(n)
integer,	intent(out)	::	info	LAPACK return code (0 = success)

private subroutine beuler_step(state)

Advance the solution by one time step using backward (implicit) Euler.

Arguments

Type	Intent	Optional	Attributes		Name
type(solver_state_t),	intent(inout)			::	state

private subroutine bdf2_step(state)

Advance the solution by one time step using the BDF2 (Gear) scheme.

Arguments

Type	Intent	Optional	Attributes		Name
type(solver_state_t),	intent(inout)			::	state

private pure subroutine pack_field(src, dst, nq, np)

Flatten a (neq x n_pt) field into a length-neq*n_pt 1-D array. Ordering: (eq=1,cell=1), (eq=2,cell=1), ..., (eq=neq,cell=n_pt).

Arguments

Type	Intent		Name
real(kind=wp),	intent(in)	::	src(nq,np)
real(kind=wp),	intent(out)	::	dst(nq*np)
integer,	intent(in)	::	nq
integer,	intent(in)	::	np

private subroutine unpack_add(src, dst, nq, np)

Add a flattened 1-D correction to a (neq x n_pt) 2-D field in place.

Arguments

Type	Intent		Name
real(kind=wp),	intent(in)	::	src(nq*np)
real(kind=wp),	intent(inout)	::	dst(nq,np)
integer,	intent(in)	::	nq
integer,	intent(in)	::	np

private pure subroutine jac_store_col(ab_loc, j_col, col, kl_loc, ku_loc, diag_row_loc, n)

Store one Jacobian column into band storage AB(diag_row_loc+i-j, j) = J(i,j). Works for both the custom (diag_row_loc = ku+1) and LAPACK (diag_row_loc = kl+ku+1) storage layouts.

Arguments

Type	Intent		Name
real(kind=wp),	intent(inout)	::	ab_loc(:,:)
integer,	intent(in)	::	j_col
real(kind=wp),	intent(in)	::	col(n)
integer,	intent(in)	::	kl_loc
integer,	intent(in)	::	ku_loc
integer,	intent(in)	::	diag_row_loc
integer,	intent(in)	::	n

private subroutine band_lu_solve(ab_loc, n, kl_loc, ku_loc, b, x)

Banded Gaussian elimination without pivoting. Band storage: AB(ku+1+i-j, j) = A(i,j). On exit x = A^{-1} b. The matrix ab_loc is overwritten with LU factors.

Arguments

Type	Intent		Name
real(kind=wp),	intent(inout)	::	ab_loc(kl_loc+ku_loc+1,n)
integer,	intent(in)	::	n
integer,	intent(in)	::	kl_loc
integer,	intent(in)	::	ku_loc
real(kind=wp),	intent(in)	::	b(n)
real(kind=wp),	intent(out)	::	x(n)

private subroutine compute_resid_glob(state)

Accumulate the global L2 norm of the residual into state%resid_glob.

Arguments

Type	Intent	Optional	Attributes		Name
type(solver_state_t),	intent(inout)			::	state

time_integration Module

Contents

Variables

Abstract Interfaces

Subroutines

Uses

Variables

Abstract Interfaces

abstract interface

public subroutine stepper_iface(state)

Arguments

Subroutines

public subroutine resolve_time_scheme(stepper, scheme)

Arguments

public subroutine init_time_scheme(scheme)

Arguments

private subroutine tvd_rk3_step(state)

Arguments

private subroutine ssprk54_step(state)

Arguments

private subroutine euler_step(state)

Arguments

private subroutine ssprk22_step(state)

Arguments

private subroutine rk4_step(state)

Arguments

private subroutine setup_band_storage(state, n_dof, kl, ku, ldab, diag_row)

Arguments

private subroutine band_lapack_solve(ab, kl, ku, n, b, info)

Arguments

private subroutine beuler_step(state)

Arguments

private subroutine bdf2_step(state)

Arguments

private pure subroutine pack_field(src, dst, nq, np)

Arguments

private subroutine unpack_add(src, dst, nq, np)

Arguments

private pure subroutine jac_store_col(ab_loc, j_col, col, kl_loc, ku_loc, diag_row_loc, n)

Arguments

private subroutine band_lu_solve(ab_loc, n, kl_loc, ku_loc, b, x)

Arguments

private subroutine compute_resid_glob(state)

Arguments