Sydney Dynamics Group Seminar: Prof. Kenneth Duru, University of Texas at El Paso -- Sydney Dynamics Group Seminar

Seminar : Sydney Dynamics group Seminar.  
Day/time : Thursday, August 6 at 4:00 PM.
Location : SMRI seminar room (A12 Macleay Room 301).  
Speaker : Kenneth Duru, University of Texas at El Paso 

Title : 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 equations (PDEs) are attractive because they are efficient on
modern and next generation hardware architectures.  However, the design of provably
stable high-order accurate numerical methods for nonlinear hyperbolic conservation laws
pose a significant challenge, as initial attempts often result in crashes due to com-
pounding 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 compress- ible 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
mo- tion 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.