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Session Overview
MS 26b: High order multi-dimensional structure preserving methods
Thursday, 15/July/2021:
4:00pm - 6:00pm

Session Chair: Christian Klingenberg
Session Chair: Wasilij Barsukow
Virtual location: Zoom 3

Session Abstract

a rich phenomenology. Numerical methods originally developed in one-dimensional situations

require excessive grid refinement in order to capture multi-dimensional phenomena, such as

vortices, differential constraints, stationary states and other structures. These structures are

themselves mostly given as differential operators.

For being efficient, it is not sufficient to merely use high order methods. They need to be

equipped with structure-preserving properties. There is a qualitative difference in the results

obtained by methods which do not preserve any structures and by those which preserve

structures. This is true even if the structures in question are differential operators, and are

preserved only as a discretization. Such methods achieve unprecedented accuracy on poorly

resolved grids. This minisymposium shall bring together researchers dealing with structure

preservation in different contexts and foster exchange on recent and trend-setting strategies to

achieve this properties.

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4:00pm - 4:30pm

An asymptotic preserving scheme for the two-dimensional shallow water equations with Coriolis forces

Alina Chertock

North Carolina State University, United States of America

We consider the two-dimensional Saint-Venant system of shallow water equations with Coriolis forces. We focus on the case of a low Froude number, in which the system is stiff and conventional explicit numerical methods are extremely inefficient and often impractical. Our goal is to design an asymptotic preserving (AP) scheme, which is uniformly asymptotically consistent and stable for a broad range of (low) Froude numbers. The goal is achieved using the flux splitting, where we split the flux into the stiff and nonstiff parts and then use an implicit-explicit approach: apply an explicit hyperbolic solver to the nonstiff part of the system while treating the stiff part of it implicitly. Moreover, the stiff part of the flux is linear and therefore we reduce the implicit stage of the proposed method to solving a Poisson-type elliptic equation, which is discretized using a standard second-order central difference scheme.

We present a series of numerical experiments, which demonstrate that the developed AP scheme achieves the theoretical second-order rate of convergence and the time-step stability restriction is independent of the Froude number. This makes the proposed AP scheme an efficient and robust alternative to fully explicit numerical methods.

4:30pm - 5:00pm

Arbitrary order structure preserving discontinuous Galerkin methods for the Euler equations with gravitation

Yulong Xing

The Ohio State University, United States of America

Hydrodynamical evolution in a gravitational field arises in many astrophysical and atmospheric problems. Improper treatment of the gravitational force can lead to a solution which oscillates around the equilibrium. In this presentation, we will talk about arbitrary order structure preserving discontinuous Galerkin finite element methods which can exactly capture the non-trivial steady state solutions of these models, and at the same time maintain the non-negativity of some physical quantities. Numerical tests are provided to verify the well-balanced property, positivity-preserving property, high-order accuracy, and good resolution for both smooth and discontinuous solutions.

5:00pm - 5:30pm

Efficient and accurate structure preserving schemes for complex nonlinear systems

Jie Shen

Purdue University, United States of America

Many complex nonlinear systems have intrinsic structures such as energy

dissipation or conservation, and/or positivity/maximum principle preserving. It is

desirable, sometimes necessary, to preserve these structures in a numerical scheme.

I will present some recent advances on using the scalar auxiliary variable (SAV) approach to develop highly efficient and accurate structure preserving schemes for a large class of complex nonlinear systems. These schemes can preserve energy dissipation/conservation as well as other global constraints and/or are positivity/bound preserving, only require solving decoupled linear equations with constant coefficients at each time step, and can achieve higher-order accuracy.

5:30pm - 6:00pm

Structure preserving methods for the compressible Euler equations with gravity

Christian Klingenberg

Wuerzburg University, Germany, Germany

We consider astrophysical systems that are modeled by the multidimensional Euler equations with gravity.

First for the homogeneous Euler equations we look at flow in the low Mach number regime. Here for conventional finite volume discretizations one has excessive dissipation in this regime. We identify inconsistent scaling for low Mach numbers of the numerical flux function as the origin of this problem. Based on the Roe solver a technique that allows to correctly represent low Mach number flows with a discretization of the compressible Euler equations is proposed. We analyze properties of this scheme and demonstrate that its limit yields a discretization of the incompressible limit system.

Next for the Euler equations with gravity we seek well-balanced methods. We describe a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the special case of stationary solutions with zero velocity, described by a nonlinear PDE, whose solutions are called hydrostatic equilibria. We present well-balanced methods, for which we can ensure robustness, accuracy and stability, since it satisfies discrete entropy inequalities.

We will then present work in progress where we combine the two methods above.

This is joint work with Fritz Röpke, Claudius Birke, Praveen Chandrashekar, Roge Käppeli and others.

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