Conservative and Stable Degree Preserving SBP Operators for Non-conforming Meshes

Abstract

Non-conforming numerical approximations offer increased flexibility for applications that require high resolution in a localized area of the computational domain or near complex geometries. Two key properties for non-conforming methods to be applicable to real world applications are conservation and energy stability. The summation-by-parts (SBP) property, which certain finite-difference and discontinuous Galerkin methods have, finds success for the numerical approximation of hyperbolic conservation laws, because the proofs of energy stability and conservation can discretely mimic the continuous analysis of partial differential equations. In addition, SBP methods can be developed with high-order accuracy, which is useful for simulations that contain multiple spatial and temporal scales. However, existing non-conforming SBP schemes result in a reduction of the overall degree of the scheme, which leads to a reduction in the order of the solution error. This loss of degree is due to the particular interface coupling through a simultaneous-approximation-term (SAT). We present in this work a novel class of SBP–SAT operators that maintain conservation, energy stability, and have no loss of the degree of the scheme for non-conforming approximations. The new degree preserving discretizations require an ansatz that the norm matrix of the SBP operator is of a degree $\ge 2p$, in contrast to, for example, existing finite difference SBP operators, where the norm matrix is $2p-1$ accurate. We demonstrate the fundamental properties of the new scheme with rigorous mathematical analysis as well as numerical verification.

Appendices

A Projection Operators

In this section we briefly discuss the construction of the projection operators $\mathsf {P}_{\xi ,L}$ and $\mathsf {P}_{\xi ,R}$ of degree p, which are applied in Sect. 5.3. Here we consider tensor-product based projection operators. As for the tensor-product SBP operators, we derive the projection operators in one dimension, denoted by $\mathsf {P}_{\xi ,L}^{(1\text {D})}$ and $\mathsf {P}_{\xi ,R}^{(1\text {D})}$. We focus on a left element L and a right element R with nodal distributions $\varvec{\eta }_L$ and $\varvec{\eta }_R$. Let $\varvec{\eta }_{{\varGamma }}$ denote the nodal distribution on the intermediate grid, then (2.16) and (2.18) give us

$$\begin{aligned} \begin{aligned} \mathsf {P}_{\xi ,L}^{(1\text {D})}\varvec{\eta }_{L}^{k}&=\varvec{\eta }_{{\varGamma }}^{k}, \\ \left( \mathsf {P}_{\xi ,L}^{(1\text {D})}\right) ^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}\varvec{\eta }_{L}^{k}&=\mathsf {H}_{\eta _{L}}^{(1\text {D})}\varvec{\eta }_{{\varGamma }}^{k}, \end{aligned} \end{aligned}$$

(A.1)

and

$$\begin{aligned} \begin{aligned} \mathsf {P}_{\xi ,R}^{(1\text {D})}\varvec{\eta }_{R}^{k}&=\varvec{\eta }_{{\varGamma }}^{k}, \\ \left( \mathsf {P}_{\xi ,R}^{(1\text {D})}\right) ^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}\varvec{\eta }_{R}^{k}&=\mathsf {H}_{\eta _{R}}^{(1\text {D})}\varvec{\eta }_{{\varGamma }}^{k}, \end{aligned} \end{aligned}$$

(A.2)

for $k=0,\dots ,p$. We reiterate, that the choice of the nodes on the intermediate grid is nearly arbitrary, the only requirement is that a 2p-accurate quadrature rule exists. In the case of $\varvec{\eta }_{{\varGamma }}=\varvec{\eta }_L$ or $\varvec{\eta }_{{\varGamma }}=\varvec{\eta }_R$, we set the projection operators to be the identity matrix. If we consider the Gauss nodes on the intermediate grid as in Sect. 5.4, the projection matrices are fully determined in the sense that no free parameters remain. In general, (A.1) and (A.2) are insufficient to fully specify $\mathsf {P}_{\xi ,L}^{(1\text {D})}$ and $\mathsf {P}_{\xi ,R}^{(1\text {D})}$. We want the approximation of the surface integral to be as close as possible to the approximated surface integral for the conforming case, in a sense that $\mathsf {H}_{\eta _{L}}^{(1\text {D})}\approx (\mathsf {P}_{\xi ,L}^{(1\text {D})})^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}\mathsf {P}_{\xi ,L}^{(1\text {D})}$ and $\mathsf {H}_{\eta _{R}}^{(1\text {D})}\approx (\mathsf {P}_{\xi ,R}^{(1\text {D})})^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}\mathsf {P}_{\xi ,R}^{(1\text {D})}$. Therefore, we use any remaining degrees of freedom to ensure that the modulus of each eigenvalue of the matrices $\mathsf {H}_{\eta _{L}}^{(1\text {D})}-(\mathsf {P}_{\xi ,L}^{(1\text {D})})^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}\mathsf {P}_{\xi ,L}^{(1\text {D})}$ and $\mathsf {H}_{\eta _{R}}^{(1\text {D})}-(\mathsf {P}_{\xi ,R}^{(1\text {D})})^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}\mathsf {P}_{\xi ,R}^{(1\text {D})}$ is as close to zero as possible.

Similarly, we motivate this optimization by analyzing the numerical flux in the left element for the upwind SAT, i.e. for $\sigma =1$. For conforming nodal distributions, this numerical flux in one dimension for the linear advection equation, with unit wave speed, reduces to

$$\begin{aligned} \varvec{f}^{*,(1\text {D})}_{L,conf}=\tilde{\alpha }\mathsf {R}_{\xi _L,N_{\xi }}^{\text {T}}\mathsf {H}_{\eta }^{(1\text {D})}\mathsf {R}_{\xi _L,N_{\xi }}\varvec{u}_{L}. \end{aligned}$$

(A.3)

For $\sigma =1$, the numerical flux of the non-conforming approximation (2.14) reduces to

$$\begin{aligned} \varvec{f}^{*,(1\text {D})}_{L}=\frac{\tilde{\alpha }}{2}\mathsf {R}_{\xi _L,N_{\xi }}^{\text {T}}\left( \mathsf {H}_{\eta _{L}}^{(1\text {D})}+\left( \mathsf {P}_{\xi ,L}^{(1\text {D})}\right) ^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}\mathsf {P}_{\xi ,L}^{(1\text {D})}\right) \mathsf {R}_{\xi _L,N_{\xi }}\varvec{u}_{L}. \end{aligned}$$

(A.4)

Here, $\varvec{f}^{*,(1\text {D})}_L=\varvec{f}^{*,(1\text {D})}_{L,conf}$, if $\mathsf {H}_{\eta _{L}}^{(1\text {D})}=(\mathsf {P}_{\xi ,L}^{(1\text {D})})^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}\mathsf {P}_{\xi ,L}^{(1\text {D})}$.

Our approach to optimize the eigenvalues of $\mathsf {H}_{\eta _{L}}^{(1\text {D})}-(\mathsf {P}_{\xi ,L}^{(1\text {D})})^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}\mathsf {P}_{\xi ,L}^{(1\text {D})}$ and $\mathsf {H}_{\eta _{R}}^{(1\text {D})}-(\mathsf {P}_{\xi ,R}^{(1\text {D})})^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}\mathsf {P}_{\xi ,R}^{(1\text {D})}$ is based on the work of Kozdon and Wilcox [20]. Here, a Levenberg–Marquardt optimization algorithm has been used. We set the number of optimization iterations within the algorithm to 10. For constructing one-dimensional projection operators a MATLAB code is provided in the electronic supplementary material (ESM).

B Comparison to the Non-conforming Method of Kozdon and Wilcox

To put the derivations using the multi-dimensional notation into context, it is useful to observe that for the tensor-product ansatz our approach is equivalent to the scheme of Kozdon and Wilcox [20]. In their notation, a projection from the left element to the glue grid is denoted by the one-dimensional projection operator $\mathsf {P}_{L2G}$. The back transformation is denoted by $\mathsf {P}_{G2L}$. If we also consider a tensor-product approximation and take the glue grid to be equivalent to our intermediate grid, then

$$\begin{aligned} \mathsf {P}_{L2G}=\mathsf {P}_{\xi ,L}^{(1\text {D})}. \end{aligned}$$

(B.1)

Due to the accuracy conditions made for $\mathsf {P}_{\xi ,L}^{(1\text {D})}$, the operator $\mathsf {P}_{L2G}$ is also p-th degree.

For the approach of Kozdon and Wilcox, as for the approach of Mattsson and Carpenter [24], the projection operators need to satisfy the following condition to ensure stability:

$$\begin{aligned} \mathsf {H}_{\eta _{L}}^{(1\text {D})}\mathsf {P}_{G2L}=\mathsf {P}_{L2G}^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}. \end{aligned}$$

(B.2)

Motivated by the stability condition one can construct a transformation back to left element

$$\begin{aligned} \mathsf {P}_{G2L}=\left( \mathsf {H}_{\eta _{L}}^{(1\text {D})}\right) ^{-1}\left( \mathsf {P}_{\xi ,L}^{(1\text {D})}\right) ^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}^{(1\text {D})}. \end{aligned}$$

(B.3)

Multiplying (B.3) by a monomial, say $\varvec{\eta }_{{\varGamma }}^{k}$ with $k=0,\dots ,p$, and using (2.18) we find

$$\begin{aligned} \mathsf {P}_{G2L}\varvec{\eta }_{{\varGamma }}^{k}=\left( \mathsf {H}_{\eta _{L}}^{(1\text {D})}\right) ^{-1}\underbrace{\left( \mathsf {P}_{\xi ,L}^{(1\text {D})}\right) ^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}\varvec{\eta }_{{\varGamma }}^{k}}_{=\mathsf {H}_{\eta _{L}}^{(1\text {D})}\varvec{\eta }_{L}^{k}}=\varvec{\eta }_{L}^{k}. \end{aligned}$$

(B.4)

So the $\mathsf {P}_{G2L}$ is also p-th accurate. Considering the symmetric SAT in the discretization (2.12 with $\sigma =0$) and focusing on tensor-product operators, we get

$$\begin{aligned} \varvec{SAT}=\frac{\tilde{\alpha }_{L}}{2\mathcal {J}_L}\mathsf {H}_{L}^{-1}\left( \mathsf {R}_{\xi _L,N_{\xi }}^{\text {T}}\mathsf {H}_{\eta _{L}}^{(1\text {D})}\mathsf {R}_{\xi _L,N_{\xi }}\varvec{u}_{L}-\mathsf {R}_{\xi _L,N_{\xi }}^{\text {T}}\left( \mathsf {P}_{\xi ,L}^{(1\text {D})}\right) ^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}\mathsf {P}_{\xi ,R}^{(1\text {D})}\mathsf {R}_{\xi _R,1}\varvec{u}_{R}\right) .\nonumber \\ \end{aligned}$$

(B.5)

Denoting $\varvec{u}_{L,N}:= \mathsf {R}_{\xi _L,N_{\xi }}\varvec{u}_{L}$ and $\varvec{u}_{R,1}:= \mathsf {R}_{\xi _{R},1}\varvec{u}_{R}$ we can rewrite (B.5) to become

$$\begin{aligned} \begin{aligned} SAT&=\frac{\tilde{\alpha }_{L}}{2\mathcal {J}_L}\left( \varvec{u}_{L,N}-\underbrace{\left( \mathsf {H}_{\eta _{L}}^{(1\text {D})}\right) ^{-1} \left( \mathsf {P}_{\xi ,L}^{(1\text {D})}\right) ^{\text {T}}\mathsf {M}_{\hat{{\varGamma }}}}_{=\mathsf {P}_{G2L}}\underbrace{\mathsf {P}_{\xi ,R}^{(1\text {D})}}_{=\mathsf {P}_{R2G}}\varvec{u}_{R,1}\right) , \\&=\frac{\tilde{\alpha }_{L}}{2\mathcal {J}_L}\left( \varvec{u}_{L,N}-\mathsf {P}_{G2L}\mathsf {P}_{R2G}\varvec{u}_{R,1}\right) , \end{aligned} \end{aligned}$$

(B.6)

which is precisely the SAT for the Kozdon and Wilcox approach, assuming a symmetric SAT, despite the fact we have a different approach to derive the SAT.