Efficient high-order time discretization methods for PDEs

May 11-13, 2022 - Villa Orlandi, Anacapri, Italy

Contributed Talk

Fully-discrete Well-Balanced order-adaptive Compact Approximate Method for Systems Balance laws

Emanuele Macca,

University of Catania, Italy

Abstract

Compact Approximate Taylor (CAT) methods for systems of conservation laws were introduced by H. Carrillo and C. Parés in 2019. These methods, based on a strategy that allows one to extend high-order Lax-Wendroff methods to nonlinear systems without using the Cauchy-Kovalevskaya procedure, have arbitrary even order of accuracy $2p$ and use ($2p+1$)-point stencils, where $p$ is an arbitrary positive integer. More recently in 2021 H. Carrillo, E. M., C. Parés, G. Russo and D. Zorío introduced a strategy to get rid of the spurious oscillations close to discontinuities produced by CAT methods. This strategy led to the so-called Adaptive CAT (ACAT) methods, in which the order of accuracy $-$ and thus the width of the stencils $-$ is adapted to the local smoothness of the solution.
A new family of well-balanced high-order shock-capturing finite difference numerical methods for systems of balance law is presented. To do this, the source term is written as the derivative of its indefinite integral that is formally treated as a flux function. The well-balanced property of the methods is discussed and a variant that allows in principle to preserve any stationary solution is presented. The resulting methods are then applied to a number of systems going from a linear scalar conservation law to the 2D Euler equations with gravity, passing by the Burgers equations with source term and the 1D shallow water equations.


back to list of speakers