Compact Approximate Taylor Methods for Systems of Conservation Laws

Abstract

A new family of high order methods for systems of conservation laws are introduced: the Compact Approximate Taylor (CAT) methods. These methods are based on centered (2p + 1)point stencils where p is an arbitrary integer. We prove that the order of accuracy is 2p and that CAT methods are an extension of high-order Lax-Wendroff methods for linear problems. Due to this, they are linearly L 2 -stable under a CF L − 1 condition. In order to prevent the spurious oscillations that appear close to discontinuities two shock-capturing techniques have been considered: a flux-limiter technique (FL-CAT methods) and WENO reconstruction for the first time derivative (WENO-CAT methods). We follow [14] in the second approach. A number of test cases are considered to compare these methods with other WENO-based schemes: the linear transport equation, Burgers equation, and the 1D compressible Euler system are considered. Although CAT methods present an extra computational cost due to the local character, this extra cost is compensated by the fact that they still give good solutions with CFL values close to 1.

Well-balanced adaptive compact approximate Taylor methods for systems of balance laws

“…Nevertheless, if (2p+1)-point differentiation formulas are used to compute spatial and temporal derivatives, the resulting method use (4p + 1)-point stencils while Lax-Wendroff methods for linear systems use (2p + 1)-point ones. In [4] a variant of these methods that use (2p + 1)-point stencils, the so-called Compact Approximated Taylor methods (CAT), was introduced. CAT methods were shown to reduce to the standard high-order Lax-Wendroff methods when applied to linear problems.…”

“…The technique used to reduce the length of the stencils increases the computational cost of a time step compared to the methods introduced in [36]: the Taylor expansions are computed locally, so that the total number of expansions needed to update the numerical solution is multiplied by (2p + 1). Nevertheless, CAT methods have better stability properties allowing larger time steps, thus compensating the extra cost per time iteration: see [4].…”

Lax Wendroff approximate Taylor methods with fast and optimized weighted essentially non-oscillatory reconstructions

“…which is different from the standard Lax-Wendroff method and whose stability properties are worse (see [18]). Compact Approximate Taylor (CAT) methods were designed in [19] as a variant of these methods that properly generalize the Lax-Wendroff methods for linear systems.…”

“…Although both LAT and CAT strategies have been combined previously with standard WENO reconstructions as equipment for attaining shock-capturing properties (see [17] and [19]), they have been never combined with FOWENO reconstructions: the goal of this work is to introduce new families of high-order numerical methods using FOWENO reconstructions and Approximate Taylor methods. These methods will be compared between them and against standard WENO implementations in a number of test cases ranging from scalar linear 1d problems to nonlinear systems of conservation laws in 2d.…”

On the Conservation Properties in Multiple Scale Coupling and Simulation for Darcy Flow with Hyperbolic-Transport in Complex Flows

“…In [27,28,58] a number of the existing admissibility (or entropy) conditions are revisited and the so-called germs that underly these conditions are identified. The seminal survey article [28] (see also [107,108,113,41,111,90,47,88]) of recent developments helps to better understand the issue of admissibility of solutions in relation with specific modeling assumptions.…”