DG_implementation – SelfForce1D

The implementation of the interfaces defined in DG_structures as well as internal routines that should never be called by a user.

Uses

- Ancestors:
- DG_structures
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Functions

JacobiP Vandermonde1D GradJacobiP GradVandermonde1D Dmatrix1D Lift1D Inverse Filter1D

Module Procedures

init_ref_element deallocate_ref_element char_flux_real char_flux_complex

Functions

function JacobiP(x, alpha, beta, n)

Function to evaluate Jacobi Polynomial $P^{(\alpha,\beta)}_n$ of type $(\alpha,\beta)>-1$ (with $\alpha+\beta\neq -1$ ) at points $x$ for order $n$ .

Arguments

Type	Intent	Attributes		Name
real(kind=wp),	intent(in),	dimension(:)	::	x	The points $x\in[-1:1]$ at which to evaluate the Jacobi Polynomial.
real(kind=wp),	intent(in)		::	alpha	The value of $\alpha$ .
real(kind=wp),	intent(in)		::	beta	The value of $\beta$ .
integer(kind=ip),	intent(in)		::	n	The order of the Jacobi polynomial.

Return Value real(kind=wp), dimension(size(x))

Returns $P^{(\alpha,\beta)}_n(x)$ .

function Vandermonde1D(n, x)

Initialize the 1D Vandermonde matrix, $\mathcal{V}_{ij}=P_j(x_i)$ for the Legendre-Gauss-Lobatto quadrature points.

Arguments

Type	Intent	Optional	Attributes		Name
integer(kind=ip),	intent(in)			::	n	The order of the Jacobi polynomials.
real(kind=wp),	intent(in),		dimension(:)	::	x	The points $x\in[-1:1]$ at which to evaluate the Vandermonde matrix.

Return Value real(kind=wp), dimension(size(x),n+1)

The return value is the Vandermonde matrix, $\mathcal{V}_{ij}$

function GradJacobiP(x, alpha, beta, n)

Evaluate the derivative of the Jacobi polynomial of type $(\alpha,\beta)>-1$ at points $x$ for order $n$ .

Arguments

Type	Intent	Attributes		Name
real(kind=wp),	intent(in),	dimension(:)	::	x	The points $x\in[-1:1]$ at which to evaluate the derivative of the Jacobi polynomial.
real(kind=wp),	intent(in)		::	alpha	The value of $\alpha$ .
real(kind=wp),	intent(in)		::	beta	The value of $\beta$ .
integer(kind=ip),	intent(in)		::	n	The order of the Jacobi polynomials.

Return Value real(kind=wp), dimension(size(x))

The return value is the derivative of the Jacobi polynomial evaluated at $x_i$ , $\left .\frac{dP_n^{(\alpha,\beta)}}{dx} \right |_{x=x_i}$

function GradVandermonde1D(n, x)

Initialize the gradient of the modal basis $j$ at $r_i$ .

Arguments

Type	Intent	Optional	Attributes		Name
integer(kind=ip),	intent(in)			::	n	The order of the Jacobi polynomials.
real(kind=wp),	intent(in),		dimension(:)	::	x	The points $x\in[-1:1]$ at which to initialize the gradient of the modal basis.

Return Value real(kind=wp), dimension(size(x),n+1)

The return value is the derivative of the modal basis, $\mathcal{V}_{r,(ij)}$ .

function Dmatrix1D(n, x, v)

Initialize the differentiation matrix for a reference element of order $n$ .

Arguments

Type	Intent	Attributes		Name
integer(kind=ip),	intent(in)		::	n	The order of the Jacobi polynomials.
real(kind=wp),	intent(in),	dimension(n+1)	::	x	The nodal points $x\in[-1:1]$ where the differentiation matrix will provide approximations to the derivative.
real(kind=wp),	intent(in),	dimension(n+1,n+1)	::	v	The Vandermonde matrix, $\mathcal{V}_{ij}$ .

Return Value real(kind=wp), dimension(n+1,n+1)

The return value is the differentiation matrix, $\mathcal{D}_r= \mathcal{V}_r \mathcal{V}^{-1}$ .

function Lift1D(n, v)

Initialize the lift matrix, $L_{ij}$ , used to compute surface integral terms in the Discontinuous Galerkin formulation.

Arguments

Type	Intent	Optional	Attributes		Name
integer(kind=ip),	intent(in)			::	n	The order of the Jacobi polynomials.
real(kind=wp),	intent(in),		dimension(n+1,n+1)	::	v	The Vandermonde matrix, $\mathcal{V}_{ij}$ .

Return Value real(kind=wp), dimension(n+1,2)

The return value is the lift matrix, $L= \mathcal{V} \mathcal{V}^{\mathrm{T}}\mathcal{E}$ , where $\mathcal{E}$ is a $n+1\times 2$ array of the form $\begin{pmatrix} 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \cdots & 0 & 1 \end{pmatrix}$

function Inverse(a)

Helper function that calculates the inverse of a matrix.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=wp),	intent(in),		dimension(:,:)	::	a	A matrix, $A$ .

Return Value real(kind=wp), dimension(size(a,1),size(a,2))

On output contains $A^{-1}$ .

function Filter1D(n, nc, s, v, invv)

Create an exponential filter matrix that can be used to filter out high-frequency noise.

Arguments

Type	Intent	Attributes		Name
integer(kind=ip),	intent(in)		::	n	The order of the element.
integer(kind=ip),	intent(in)		::	nc	The cutoff, below which the low modes are left untouched.
integer(kind=ip),	intent(in)		::	s	The order of the filter.
real(kind=wp),	intent(in),	dimension(:,:)	::	v	The Vandermonde matrix, $\mathcal{V}$ .
real(kind=wp),	intent(in),	dimension(:,:)	::	invv	The inverse of the Vandermonde matric, $\mathcal{V}^{-1}$ .

Return Value real(kind=wp), dimension(n+1,n+1)

The return value is the filter matrix, $\mathcal{F}$ .

Subroutines

subroutine JacobiGQ(alpha, beta, n, x, w)

Compute the n'th order Gauss quadrature points, $x$ and weights, $w$ , associated with the Jacobi polynomial of type $(\alpha,\beta)>-1$ .

Arguments

Type	Intent	Attributes		Name
real(kind=wp),	intent(in)		::	alpha	The value of $\alpha$ .
real(kind=wp),	intent(in)		::	beta	The value of $\beta$ .
integer(kind=ip),	intent(in)		::	n	The order of the Jacobi polynomial.
real(kind=wp),	intent(out),	dimension(n+1)	::	x	On output the location of the Gauss quadrature points.
real(kind=wp),	intent(out),	dimension(n+1)	::	w	On output the weights.

subroutine JacobiGL(alpha, beta, n, r)

Compute the n'th order Gauss Lobatto quadrature points, $x$ , associated with the Jacobi polynomial of type $(\alpha,\beta)>-1$ .

Arguments

Type	Intent	Attributes		Name
real(kind=wp),	intent(in)		::	alpha	The value of $\alpha$ .
real(kind=wp),	intent(in)		::	beta	The value of $\beta$ .
integer(kind=ip),	intent(in)		::	n	The order of the Jacobi polynomial.
real(kind=wp),	intent(out),	dimension(n+1)	::	r	On output the location of the Gauss Lobatto quadrature points.

subroutine GaussWeigths(alpha, beta, n, r, w)

Routine to calculate the integration weights for the Gauss-Lobatto Quadrature points.

Arguments

Type	Intent	Attributes		Name
real(kind=wp),	intent(in)		::	alpha	The value of $\alpha$ .
real(kind=wp),	intent(in)		::	beta	The value of $\beta$ .
integer(kind=ip),	intent(in)		::	n	The order of the Jacobi polynomials.
real(kind=wp),	intent(in),	dimension(:)	::	r	The points for which the weights should be determined.
real(kind=wp),	intent(out),	dimension(:)	::	w	On return the integration weights.

subroutine Jacobian(n, vx, r, dr, x, rx)

Calculate the Jacobian for transforming derivatives from the reference element to the physical element.

Arguments

Type	Intent	Attributes		Name
integer(kind=ip),	intent(in)		::	n	The order of the element.
real(kind=wp),	intent(in),	dimension(2)	::	vx	A 1d real array of size 2 that contains the physical coordinates at the boundaries of the element.`
real(kind=wp),	intent(in),	dimension(n+1)	::	r	The node locations within the reference element, $r_i$ .
real(kind=wp),	intent(in),	dimension(:,:)	::	dr	The derivative matrix for the reference element, $\mathcal{D}_r$ .
real(kind=wp),	intent(out),	dimension(n+1)	::	x	On output contains the physical coordinates for this physical element, $x_i$ .
real(kind=wp),	intent(out),	dimension(n+1)	::	rx	On output contains (\frac{d r_i}{d x_j}.

Module Procedures

module procedure init_ref_element module function init_ref_element(order, sorder, nc)

Arguments

Type	Intent	Optional		Name
integer(kind=ip),	intent(in)		::	order	The order of the reference element.
integer(kind=ip),		optional	::	sorder	The order of a smoothing filter. Both this and nc have to be present before a filter is constructed.
integer(kind=ip),		optional	::	nc	No filtering of orders below this value. Both this and sorder have to be present before a filter is constructed.

Return Value type(ref_element)

The return type has to be a reference element.

module procedure deallocate_ref_element module subroutine deallocate_ref_element(refel)

Arguments

Type	Intent	Optional	Attributes		Name
type(ref_element)				::	refel	The argument has to be a reference element type.

module procedure char_flux_real module function char_flux_real(this, nvar, order, uint, uext, flux, lambda, s, sinv, debug_output)

Arguments

Type	Intent	Attributes		Name
class(ref_element),	intent(in)		::	this	Has to be a reference element type.
integer(kind=ip),	intent(in)		::	nvar	The number of variables the characteristic flux has to be computed for.
integer(kind=ip),	intent(in)		::	order	The order of the reference element. Used to declare the size of the return array.
real(kind=wp),	intent(in),	dimension(2,nvar)	::	uint	The boundary data internal to this element.
real(kind=wp),	intent(in),	dimension(2,nvar)	::	uext	The boundary data external to this element.
real(kind=wp),	intent(in),	dimension(2,nvar)	::	flux	The boundary fluxes.
real(kind=wp),	intent(in),	dimension(2,nvar)	::	lambda	The boundary characteristic speeds.
real(kind=wp),	intent(in),	dimension(2,nvar,nvar)	::	s	The matrix that converts from characteristic to evolved variables.
real(kind=wp),	intent(in),	dimension(2,nvar,nvar)	::	sinv	The matrix that converts from evolved to characteristic variables.
logical,	intent(in)		::	debug_output	Currently ignored for the real version of this routine.

Return Value real(kind=wp), dimension(order+1,nvar)

At return the numerical flux has been lifted to the whole element.

module procedure char_flux_complex module function char_flux_complex(this, nvar, order, uint, uext, flux, lambda, s, sinv, debug_output)

Arguments

Type	Intent	Attributes		Name
class(ref_element),	intent(in)		::	this	Has to be a reference element type.
integer(kind=ip),	intent(in)		::	nvar	The number of variables the characteristic flux has to be computed for.
integer(kind=ip),	intent(in)		::	order	The order of the reference element. Used to declare the size of the return array.
complex(kind=wp),	intent(in),	dimension(2,nvar)	::	uint	The boundary data internal to this element.
complex(kind=wp),	intent(in),	dimension(2,nvar)	::	uext	The boundary data external to this element.
complex(kind=wp),	intent(in),	dimension(2,nvar)	::	flux	The boundary fluxes.
real(kind=wp),	intent(in),	dimension(2,nvar)	::	lambda	The boundary characteristic speeds.
real(kind=wp),	intent(in),	dimension(2,nvar,nvar)	::	s	The matrix that converts from characteristic to evolved variables.
real(kind=wp),	intent(in),	dimension(2,nvar,nvar)	::	sinv	The matrix that converts from evolved to characteristic variables.
logical,	intent(in)		::	debug_output	If .true., produce debug output

Return Value complex(kind=wp), dimension(order+1,nvar)

At return the numerical flux has been lifted to the whole element.

DG_implementation Submodule

Contents

Functions

Subroutines

Module Procedures

Uses

Ancestors:

Contents

Functions

Subroutines

Module Procedures

Functions

function JacobiP(x, alpha, beta, n)

Arguments

Return Value real(kind=wp), dimension(size(x))

function Vandermonde1D(n, x)

Arguments

Return Value real(kind=wp), dimension(size(x),n+1)

function GradJacobiP(x, alpha, beta, n)

Arguments

Return Value real(kind=wp), dimension(size(x))

function GradVandermonde1D(n, x)

Arguments

Return Value real(kind=wp), dimension(size(x),n+1)

function Dmatrix1D(n, x, v)

Arguments

Return Value real(kind=wp), dimension(n+1,n+1)

function Lift1D(n, v)

Arguments

Return Value real(kind=wp), dimension(n+1,2)

function Inverse(a)

Arguments

Return Value real(kind=wp), dimension(size(a,1),size(a,2))

function Filter1D(n, nc, s, v, invv)

Arguments

Return Value real(kind=wp), dimension(n+1,n+1)

Subroutines

subroutine JacobiGQ(alpha, beta, n, x, w)

Arguments

subroutine JacobiGL(alpha, beta, n, r)

Arguments

subroutine GaussWeigths(alpha, beta, n, r, w)

Arguments

subroutine Jacobian(n, vx, r, dr, x, rx)

Arguments

Module Procedures

module procedure init_ref_element module function init_ref_element(order, sorder, nc) Interface →

Arguments

Return Value type(ref_element)

module procedure deallocate_ref_element module subroutine deallocate_ref_element(refel) Interface →

Arguments

module procedure char_flux_real module function char_flux_real(this, nvar, order, uint, uext, flux, lambda, s, sinv, debug_output) Interface →

Arguments

Return Value real(kind=wp), dimension(order+1,nvar)

module procedure char_flux_complex module function char_flux_complex(this, nvar, order, uint, uext, flux, lambda, s, sinv, debug_output) Interface →

Arguments

Return Value complex(kind=wp), dimension(order+1,nvar)

module procedure init_ref_element module function init_ref_element(order, sorder, nc)

module procedure deallocate_ref_element module subroutine deallocate_ref_element(refel)

module procedure char_flux_real module function char_flux_real(this, nvar, order, uint, uext, flux, lambda, s, sinv, debug_output)

module procedure char_flux_complex module function char_flux_complex(this, nvar, order, uint, uext, flux, lambda, s, sinv, debug_output)