Conservative flux-differencing SBP method for nonlinear systems

Let $d\ge 2$, let ${\mathcal{T}}_h$ be a finite matching affine simplicial mesh, and consider $U_t+{\sum }_{\mathrm{\ell }=1}^d{\mathrm{\partial }}_{x_{\mathrm{\ell }}}{F}_{\mathrm{\ell }}(U)=0$ with state domain $\mathcal{A}\subseteq {\mathbb{R}}^m$. Assume a common convex entropy $\eta$ and, for each direction $\mathrm{\ell }$, an entropy flux $q_{\mathrm{\ell }}$ so that $(\eta ,q_{\mathrm{\ell }})$ satisfies Convex entropy pairs, entropy variables, and entropy potentials for systems for the directional flux $F_{\mathrm{\ell }}$. Write $v=\mathrm{\nabla }\eta$ and ${\psi }_{\mathrm{\ell }}=v^T{F}_{\mathrm{\ell }}-q_{\mathrm{\ell }}$. For every direction, supply a symmetric consistent two-point flux $F_{\mathrm{\ell }}^{ec}$ satisfying Tadmor entropy-conservative two-point flux, namely $(v(U_R)-v(U_L){)}^T{F}_{\mathrm{\ell }}^{ec}(U_L,U_R)={\psi }_{\mathrm{\ell }}(U_R)-{\psi }_{\mathrm{\ell }}(U_L)\mathrm{.}$

On each cell use diagonal-norm SBP data from Diagonal-norm summation-by-parts element data. Put $Q_{K,\mathrm{\ell }}=H_K{D}_{K,\mathrm{\ell }}$; let $U_i^K\in \mathcal{A}$ be the nodal states; and let the face restrictions select paired face nodes with matching positive weights on the two incident cells. For an oriented interior face node with normal $n$ from the minus trace to the plus trace, set $F_n(U)={\sum }_{\mathrm{\ell }}{n}_{\mathrm{\ell }}{F}_{\mathrm{\ell }}(U)$ and ${\psi }_n(U)={\sum }_{\mathrm{\ell }}{n}_{\mathrm{\ell }}{\psi }_{\mathrm{\ell }}(U)$. Given any fixed coefficient ${\alpha }_{F,q}\ge 0$, define the normal interface flux ${\widehat{F}}_{n,F,q}^{\ast }(U^{-},U^{+})={\sum }_{\mathrm{\ell }=1}^d{n}_{\mathrm{\ell }}{F}_{\mathrm{\ell }}^{ec}(U^{-},U^{+})-\frac{{\alpha }_{F,q}}{2}(v(U^{+})-v(U^{-}))\mathrm{.}$ The coefficient is the same when the face orientation is reversed. Thus the flux is conservative and consistent, and its entropy contraction is $[v{]}^T{\widehat{F}}_{n,F,q}^{\ast }=[{\psi }_n]-\frac{{\alpha }_{F,q}}{2}\parallel [v]{\parallel }^2\le [{\psi }_n],$ by the directional Tadmor identities in Tadmor entropy-conservative two-point flux. On physical boundary faces supply a consistent normal flux and require the boundary entropy contribution below to be nonnegative.

Write ${\omega }_{F,q}$ for the $q$th diagonal entry of $M_F$. Define the boundary entropy contribution ${\mathcal{B}}_h(t)={\sum }_{F\subset \mathrm{\partial }\mathrm{\Omega }}{\sum }_q{\omega }_{F,q}(v(U_{F,q}{)}^T{\widehat{F}}_{n_F}^{\ast }-{\psi }_{n_F}(U_{F,q}))\mathrm{.}$ The selected boundary flux is entropy admissible when ${\mathcal{B}}_h(t)\ge 0$. The flux-differencing SBP semidiscretization on each element is $H_K{\dot{U}}^K+2{\sum }_{\mathrm{\ell }=1}^d(Q_{K,\mathrm{\ell }}\circ F_{\mathrm{\ell }}^{ec}(U^K,U^K))\mathbf{1}+{\sum }_{F\subset \mathrm{\partial }K}{R}_{K,F}^T{M}_{K,F}({\widehat{F}}_{n_{K,F}}^{\ast }-F_{n_{K,F}}(U_{K,F}))=0,$ where $\circ$ is entrywise matrix multiplication and $F_{\mathrm{\ell }}^{ec}(U^K,U^K)$ is the matrix with $(i,j)$ entry $F_{\mathrm{\ell }}^{ec}(U_i^K,U_j^K)$. These are conditional method data: compatible SBP data and directional entropy-conservative fluxes are hypotheses, not asserted to exist for every mesh or entropy pair. The interior numerical flux is explicitly defined once those data and ${\alpha }_{F,q}$ are supplied. Curved elements are excluded.