On the robustness and structure-preservation of high-order methods for nonlinear conservation laws

Abstract: Robust and effective high-order accurate numerical methods for solving partial differential equationsare attractive because they are efficient on modern and next generation hardware architectures. However, the design of provably stable numerical methods for nonlinear hyperbolic conservation laws pose a significant challenge, as initial attempts often result in crashes due to compounding numerical errors or the presence of undesirable numerical artifacts which can pollute numerical simulations everywhere. Desirable high order accurate methods for nonlinear PDEs must be robust (provably stable) and preserve several important invariants present in the system.

The key strategy for developing robust high-order methods for nonlinear conservation laws is to design the numerical methods to as far as possible emulate the entropy-stability properties of the continuous model at the discrete level. However, to succeed, the system of nonlinear conservation laws must be rewritten into the so-called skew-symmetric or entropy-conserving split-form so that entropy and energy analysis use only integration by parts and forgoes the use of the chain rule and product rule at the discrete level. Furthermore, other than the primary motivation of showing entropy-stability, it is also desirable that the skew-symmetric reformulation ensures structure preservation using only integration by parts. For example research on total energy conserving discretizations is critical in order to ensure discrete energy balance and improve numerical simulation results for Earth system models. In particular for atmospheric flow problems it is desirable that numerical methods preserve: vorticity dynamics, geostrophic balance, mass, energy, entropy, buoyancy, tracer-variance, and thermodynamic consistency. 

In this presentation our objectives are two-fold: One, we will identify suitable mathematical entropy pairs to prove entropy-stability for the nonlinear thermal shallow water equation and the moist compressible Euler equation, formulate structure preserving coordinate transformations, and perform mimetic reformulations in complex geometries. Our skew-symmetric reformulations ensure well-posedness of the models and guarantee structure preservation. More importantly, our reformulations of equations of motion can be targeted by summation-by-parts (SBP) discretizations which enable provably entropy-stable numerical approximations and ensure discrete structure preservation.

Two, we will present the dual-pairing (DP) and upwind SBP framework for accurate and robust numerical approximations of nonlinear conservation laws. As opposed to conventional discontinuous Galerkin methods which can only induce dissipation through numerical fluxes acting at element interfaces, the DP SBP are designed to be upwind, that is they come with an inbuilt ”filters” of which its goal is to detect and effectively resolve regions where the solution is poorly resolved and/or discontinuities are found, while maintaining high-order accuracy and numerical stability. Numerical experiments are presented to verify accuracy and demonstrate the robustness of our numerical framework.