## Summation By Parts

Date

Abstract

Calculate ﬁrst derivates of grid functions using ﬁnite diﬀerence stencils that satisfy summation by parts.

### 1 Introduction

Given a discretization ${x}_{0}\dots {x}_{N}$ of a computational domain $x\in \left[a,b\right]$ with gridspacing $h$ a one dimensional ﬁnite diﬀerence operator approximation to a ﬁrst derivative, $D$, is said to satisfy summation by parts (SBP) with respect to a scalar product (deﬁned by its coeﬃcients ${\sigma }_{ij}$)

 ${⟨u,v⟩}_{h}=h\sum _{i=0}^{N}{u}_{i}{v}_{j}{\sigma }_{ij}$ (1)

if the property

 ${⟨u,Dv⟩}_{h}+{⟨Du,v⟩}_{h}={\left(uv\right)|}_{a}^{b}$ (2)

is satisﬁed for all possible gridfunctions $u$ and $v$.

At a given ﬁnite diﬀerence order, there are several diﬀerent ways of doing this depending on the structure of the scalar product. The three commonly considered cases are the diagonal norm, the restricted full norm and the full norm (see ﬁgure 1 for the structure).

Figure 1: The structure of the scalar product matrix in the diagonal case (left), the restricted full case (middle) and the full case (right) for the 4th order interior operators. Only non-zero elements are shown.

In the following we denote the order of accuracy at the boundary by $\tau$, the order in the interior by $s$ and the width of the boundary region1 by $r$.

For the diagonal norm case it turns out that with $r=2\tau$ it is possible to ﬁnd SBP operators with $s=2\tau$ (at least when $\tau \le 4$), i.e. the order of accuracy at the boundary is half the order in the interior. For the restricted full norm case $\tau =s-1$ when $r=\tau +2$ and for the full norm case $\tau =s-1$ when $r=\tau +1$.

The operators are named after their norm and their interior and boundary orders. Thus, for example, we talk about diagonal norm 6-3 operators and restricted full norm 4-3 operators.

In the diagonal case the 2-1 and 4-3 operators are unique whereas the 6-3 and 8-4 operators have 1 and 3 free parameters, respectively. In the restricted full norm case the 4-3 operators have 3 free parameters and the 6-5 operators have 4, while in the full norm case the number of free parameters is 1 less than the restricted full case.

### 2 Numerical Implementation

Currently this thorn implements only diagonal and restricted full norm SBP operators. The diagonal norm 2-1, 4-2 and 6-3 and the restricted full norm 4-3 are the ones listed in [1] where in the presence of free parameters the set of parameters giving a minimal bandwidth have been chosen. For the diagonal norm 8-4 case the minimal bandwidth choice are to restrictive with respect to the Courant factor and in this case parameters are chosen so as to maximize the Courant factor2 . In addition the restricted full norm 6-5 operator has been implemented. This was calculated using a Mathematica script kindly provided by José M. Martín-García.

For the restricted full norm 6-5 SBP operators a compatible dissipation operator has been implemented as well based on [3].

### 3 Using This Thorn

The basic functionality of calculating ﬁnite diﬀerence approximations are performed by a set of aliased functions. In this way things are kept ﬂexible in the sense that diﬀerent thorns each can provide their own way of calculating derivatives but if they use the same calling interface the user can switch between diﬀerent derivative schemes by simply activating the appropriate ﬁnite diﬀerencing thorn. Also the same derivative routines can be called from both C and Fortran.

There are currently two diﬀerent calling interfaces corresponding to diﬀerent levels of ﬂexibility. In the ﬁrst case the fact that derivatives are approximated by ﬁnite diﬀerencing is completely hidden for the user. Using this interface the same physics thorn could in principle be used with a ﬁnite diﬀerence driver, a ﬁnite elements driver or a spectral driver. There is however a price to pay for this ﬂexibility due to the fact that the operations are performed on whole grid functions at a time so that storage for the derivatives has to be provided. This can signiﬁcantly increase the amount of memery required for the evolution. The alternative interface returns instead the ﬁnite diﬀerence coeﬃcients and allows the user to calculate derivatives on a pointwise basis. This can signiﬁcantly save memory but limits the physics thorn to use ﬁnite diﬀerences.

#### 3.1 Obtaining This Thorn

The thorn is currently still in development and so is not generally available. Access can be requested by contacting the thorn maintainer.

#### 3.2 Basic Usage

In order to use the derivative routines the appropriate declarations have to be added to your thorns interface.ccl ﬁle. To use the interface that works on whole grid functions you have to add:

SUBROUTINE Diff_gv ( CCTK_POINTER_TO_CONST IN cctkGH, \
CCTK_INT IN dir, \
CCTK_REAL IN ARRAY var, \
CCTK_REAL OUT ARRAY dvar, \
CCTK_INT IN table_handle )
USES FUNCTION Diff_gv

Here cctkGH is the pointer to the cactus GH, dir is the direction of the derivative (0 for $x$, 1 for $y$ and 2 for $z$), var is the grid function to calculate derivatives of and dvar is the grid function where the derivative is to be stored and table_handle is an integer that contains a handle for a keyword table with optional parameters. If there are no optional paramters just pass in a negative value.

To use the interface that returns the ﬁnite diﬀerence coeﬃcients you have to add:

SUBROUTINE Diff_coeff ( CCTK_POINTER_TO_CONST IN cctkGH, \
CCTK_INT IN dir, \
CCTK_INT IN nsize, \
CCTK_INT OUT ARRAY imin, \
CCTK_INT OUT ARRAY imax, \
CCTK_REAL OUT ARRAY q, \
CCTK_INT IN table_handle )

Here, again, cctkGH is the pointer to the cactus GH, dir is the direction of the derivative, nsize is the grid size in the direction you are interested in, imin and imax are 1D integer arrays of size nsize that on return will contain the minimum and maximum index of the stencil in direction dir, q is a 2D real array of size nsize$×$nsize that on return will contain the coeﬃcients for the derivatives. The table_handle serves the same purpose as in the previous case.

#### 3.3 Examples

The following piece of Fortran code, shows how to calculate derivatives of a grid function, f, and store the derivatives in the x-, y- and z-directions in the grid functions dxf, dyf and dzf:

call Diff_gv (cctkGH, 0, f, dxf, -1)
call Diff_gv (cctkGH, 1, f, dyf, -1)
call Diff_gv (cctkGH, 2, f, dzf, -1)

In order to use the interface for doing the pointwise derivatives, the following Fortran90 example shows the necessary declarations and an example of how to calculate the derivatives:

CCTK_REAL, dimension(:,:), allocatable :: qx, qy, qz
CCTK_INT, dimension(:), allocatable :: iminx, imaxx, iminy, &
imaxy, iminz, imaxz
CCTK_REAL :: idelx, idely, idelz
CCTK_INT :: i, j, k

allocate ( qx(cctk_lsh(1),cctk_lsh(1)), &
qy(cctk_lsh(2),cctk_lsh(2)), &
qz(cctk_lsh(3),cctk_lsh(3)), &
iminx(cctk_lsh(1)), imaxx(cctk_lsh(1)), &
iminy(cctk_lsh(2)), imaxy(cctk_lsh(2)), &
iminz(cctk_lsh(3)), imaxz(cctk_lsh(3)) )

call Diff_Coeff ( cctkGH, 0, cctk_lsh(1), iminx, imaxx, qx, -1 )
call Diff_Coeff ( cctkGH, 1, cctk_lsh(2), iminy, imaxy, qy, -1 )
call Diff_Coeff ( cctkGH, 2, cctk_lsh(3), iminz, imaxz, qz, -1 )

idelx = 1.0d0 / CCTK_DELTA_SPACE(1)
idely = 1.0d0 / CCTK_DELTA_SPACE(2)
idelz = 1.0d0 / CCTK_DELTA_SPACE(3)

do k = 1, cctk_lsh(3)
do j = 1, cctk_lsh(2)
do i = 1, cctk_lsh(1)
dxf(i,j,k) = idelx * sum ( qx(iminx(i):imaxx(i),i) * &
f(iminx(i):imaxx(i),j,k) )
dyf(i,j,k) = idely * sum ( qy(iminy(j):imaxy(j),j) * &
f(i,iminy(j):imaxy(j),k) )
dzf(i,j,k) = idelz * sum ( qz(iminz(k):imaxz(k),k) * &
f(i,j,iminz(k):imaxz(k)) )

end do
end do
end do

deallocate ( qx, qy, qz, iminx, imaxx, iminy, imaxy, iminz, imaxz )

#### 3.4 Support and Feedback

This thorn is maintained by Peter Diener. Any questions and comments should be directed by e-mail to diener@cct.lsu.edu.

### 4 History

This thorn grew out of the needs of a multipatch relativity code and as such was initially designed to those needs. The addition of the possibility of passing in a handle for a keyword table was done at the request of Jonathan Thornburg and should make it easy to extend the thorn with additional features. The addition of the coeﬃcient interface was done at the request of Bela Szilagyi who felt that the storage overhead was to large for his code. In the near future the thorn will be extended with second derivatives SBP operators.

### 5 Acknowledgements

We thank José M. Martín-García for very kindly providing us with a very well designed mathematica script to calculate the ﬁnite diﬀerence and scalar product coeﬃcients.

### References

[1]   Bo Strand, 1994, Journal of Computational Physics, 110, 47–67.

[2]   Luis Lehner, Oscar Reula and Manuel Tiglio, in preparation.

[3]   Ken Mattsson, Magnus Svärd and Jan Nordström, 2004, Journal of Scientiﬁc Computing, 21, 57–79.

### 6 Parameters

 check_grid_sizes Scope: restricted BOOLEAN Description: Should we check grid sizes and ghost zones Default: yes

 diss_fraction Scope: restricted REAL Description: Fractional size of the transition region for the full restricted dissipation operator Range Default: 0.2 0:0.5

 dissipation_type Scope: restricted KEYWORD Description: Type of dissipation operator Range Default: Mattson-Svard-Nordstrom see [1] below Mattson, Svaerd and Nordstroem type Kreiss-Oliger Kreiss-Oliger modiﬁed near the boundaries

[1]

Mattson-Svard-Nordstrom

 epsdis Scope: restricted REAL Description: Dissipation strength Range Default: 0.2 *:* Values typical between 0 and 1

 h_scaling Scope: restricted REAL Description: Scaling factor for the local grid spacing in the dissipation operators Range Default: 1.0 0:* Positive please

 norm_type Scope: restricted KEYWORD Description: Type of norm Range Default: Diagonal Diagonal Diagonal norm Full restricted Full restricted norm

 onesided_interpatch_boundaries Scope: restricted BOOLEAN Description: Evaluate derivatives near the local grid boundary if it is an inter-patch boundary Default: yes

 onesided_outer_boundaries Scope: restricted BOOLEAN Description: Evaluate derivatives within ghost zones of the outer boundary Default: yes

 operator_type Scope: restricted KEYWORD Description: Type of operator Range Default: Optimized Minimal Bandwidth Minimal bandwidth (except for 8-4 which is minimal spectral radius) Optimized Optimized for performance

 order Scope: restricted INT Description: Order of accuracy Range Default: 2 2:8:2

 poison_derivatives Scope: restricted BOOLEAN Description: Should we poison Dvar at boundary_shiftout perimeter when taking derivatives Default: no

 poison_dissipation Scope: restricted BOOLEAN Description: Should we poison rhs at boundary_shiftout perimeter when applying dissipation Default: no

 poison_value Scope: restricted REAL Description: Degree of intoxication Range Default: 666.0 *:* Anything you want

 sbp_1st_deriv Scope: restricted BOOLEAN Description: Should the 1st derivative operator be SBP Default: yes

 sbp_2nd_deriv Scope: restricted BOOLEAN Description: Should the 2nd derivative operator be SBP Default: yes

 sbp_upwind_deriv Scope: restricted BOOLEAN Description: Should the upwind derivative operator be SBP Default: yes

 scale_with_h Scope: restricted BOOLEAN Description: Should we scale the dissipation with the grid spacing h Default: no

 use_dissipation Scope: restricted BOOLEAN Description: Should we add dissipation Default: no

 use_shiftout Scope: restricted BOOLEAN Description: Should we use the boundary_shift_out parameters from CoordBase to shift the stencils of derivatives and dissipation Default: no

 use_variable_deltas Scope: restricted BOOLEAN Description: Use extra grid functions to allow for variable delta’s in the dissipation operators Default: no

 vars Scope: restricted STRING Description: List of evolved grid functions that should have dissipation added Range Default: (none) .* Must be a valid list of grid functions

 zero_derivs_y Scope: restricted BOOLEAN Description: set all derivatives to 0 in the y-direction Default: no

 zero_derivs_z Scope: restricted BOOLEAN Description: set all derivatives to 0 in the z-direction Default: no

### 7 Interfaces

Implements:

summationbyparts

#### Grid Variables

##### 7.0.1 PUBLIC GROUPS
 Group Names Variable Names Details normmask compact 0 nmask description Mask for the norm calculation dimensions 3 distribution DEFAULT group type GF tags tensortypealias=”scalar” Prolongation=”None” checkpoint=”no” timelevels 1 variable type REAL deltas compact 0 sbp_dx description Dissipation deltas sbp_dy dimensions 3 sbp_dz distribution DEFAULT group type GF tags tensortypealias=”U” Prolongation=”None” timelevels 1 variable type REAL

Provides:

Diﬀ_gf to

Diﬀ_gv to

Diﬀ_up_gv to

Diﬀ2_gv to

Diﬀ_coeﬀ to

Diﬀ_up_coeﬀ to

Diﬀ2_coeﬀ to

GetScalProdDiag to

GetScalProdCoeﬀ to

GetLshIndexRanges to

GetBoundWidth to

### 8 Schedule

This section lists all the variables which are assigned storage by thorn CactusNumerical/SummationByParts. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

#### Scheduled Functions

CCTK_BASEGRID (conditional)

setup the mask for the calculation of the norm

 Language: fortran Type: function

CCTK_BASEGRID (conditional)

sbp_deltainitial

initialize dissipation deltas

 Language: fortran Type: function

CCTK_POSTINITIAL (conditional)

sbp_checkgridsizes

check grid sizes and ghost zones

 Language: fortran Type: function

MoL_PostRHS (conditional)