Simultaneous Approximation Terms for Multi-dimensional Summation-by-Parts Operators

Abstract

This paper is concerned with the accurate, conservative, and stable imposition of boundary conditions and inter-element coupling for multi-dimensional summation-by-parts (SBP) finite-difference operators. More precisely, the focus is on diagonal-norm SBP operators that are not based on tensor products and are applicable to unstructured grids composed of arbitrary elements. We show how penalty terms—simultaneous approximation terms (SATs)—can be adapted to discretizations based on multi-dimensional SBP operators to enforce boundary and interface conditions. A general SAT framework is presented that leads to conservative and stable discretizations of the variable-coefficient advection equation. This framework includes the case where there are no nodes on the boundary of the SBP element at which to apply penalties directly. This is an important generalization, because elements analogous to Legendre–Gauss collocation, i.e. without boundary nodes, typically have higher accuracy for the same number of degrees of freedom. Symmetric and upwind examples of the general SAT framework are created using a decomposition of the symmetric part of an SBP operator; these particular SATs enable the pointwise imposition of boundary and inter-element conditions. We illustrate the proposed SATs using triangular-element SBP operators with and without nodes that lie on the boundary. The accuracy, conservation, and stability properties of the resulting SBP–SAT discretizations are verified using linear advection problems with spatially varying divergence-free velocity fields.

Appendix: Satisfaction of the Discrete Divergence-Free Equation

In general, the analytical velocity, which we will denote here as $\hat{\varvec{\lambda }}_{\xi }$, does not satisfy the discretized divergence-free condition, (29). Therefore, we seek a discrete vector field that satisfies the discrete divergence-free condition and is as close as possible, in some norm, to the analytical field. This appendix describes how find such a discrete vector field.

First we solve for the face-normal velocities, $(\lambda _n)_{i} = \left( \lambda _{\xi } n_{\xi } + \lambda _{\eta } n_{\eta }\right) _{i}$, that appear in the elements of the $\mathsf {B}_{\lambda }$ matrices. A constraint on the $(\lambda _n)_{i}$ for each element is obtained by substituting $\varvec{v} = \varvec{1}$ into the identity (30):

$$\begin{aligned} \sum _{j=1}^{N_{\hat{\varGamma }}} \left( \varvec{1}_{\hat{\varGamma }_{j}}^{{{\mathrm {T}}}}\mathsf {B}_{\lambda ,j} \mathsf {R}_{j}\right) \varvec{1} = \sum _{j=1}^{N_{\hat{\varGamma }}} \sum _{i=1}^{n_{j}} b_{i}^{(j)} \left( \lambda _{n}^{(j)}\right) _{i} = \varvec{1}^{T}\left( \mathsf {\Lambda }_{\xi }\mathsf {Q}_{\xi }+ \mathsf {\Lambda }_{\eta }\mathsf {Q}_{\eta }\right) \varvec{1} = 0, \end{aligned}$$

where the last equality follows from $\mathsf {Q}_{\xi }\varvec{1} = \mathsf {Q}_{\eta }\varvec{1} = \varvec{0}$. This constraint is simply a discretization of $\int _{\hat{\varGamma }} \varvec{\lambda }\cdot \varvec{n}\, {{\mathrm {d}}}\hat{\varGamma }= 0$ on each element. There are fewer elements than face-normal velocities, so we solve a quadratic optimization problem that minimizes the Cartesian norm between the discrete and analytical values at the face nodes, $(\lambda _{n}^{(j)})_{i}$ and $(\hat{\lambda }_{n}^{(j)})_{i}$, respectively, subject to the above constraint.

Once the $\mathsf {B}_{\lambda }$ matrices are determined, we solve for the diagonal matrices $\Lambda _{\xi }$ and $\Lambda _{\eta }$. We follow a procedure analogous to the one used for $\mathsf {B}_{\lambda }$; in this case we minimize the Cartesian norm between the discrete and analytical values at the SBP nodes and (29) becomes the constraint. The optimization problems on each element are decoupled.

For the cases considered here, we verified that the $L^2$ error in the discrete velocity field is at least an order of magnitude smaller than the $L^2$ error in the scalar field ${\mathcal {U}}$. Moreover, the error in the velocity field decreases with h, the average mesh spacing, at a faster rate than the error in $\varvec{u}$, and the error in the velocity field has an insignificant impact on the solution error.