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

The goal of this work is to introduce new families of shock-capturing high-order numerical methods for systems of conservation laws that combine Fast WENO (FWENO) and Optimal WENO (OWENO) reconstructions with Approximate Taylor methods for the time discretization. FWENO reconstructions are based on smoothness indicators that require a lower number of calculations than the standard ones. OWENO reconstructions are based on a definition of the nonlinear weights that allows one to unconditionally attain the optimal order of accuracy regardless of the order of critical points. Approximate Taylor methods update the numerical solutions by using a Taylor expansion in time in which, instead of using the Cauchy-Kovalevskaya procedure, the time derivatives are computed by combining spatial and temporal numerical differentiation with Taylor expansions in a recursive way. These new methods are compared between them and against methods based on standard WENO implementations and/or SSP-RK time discretization. A number of test cases are considered ranging from scalar linear 1d problems to nonlinear systems of conservation laws in 2d

Compact Approximate Taylor methods for systems of conservation laws

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 L2-stable under a CFL<1 condition. In order to prevent the spurious oscillations that appear close to discontinuities two shock-capturing techniques have been considered: a fux-limiter technique (FL-CAT methods) and WENO reconstruction for the frst time derivative (WENO-CAT methods). We follow the WENO-Lax Wendroff Approximate Taylor method of Zorio, Baeza and Mullet (2017) 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.Comment: 10 figures, 33 page

An order-adaptive compact approximation Taylor method for systems of conservation laws

We present a new family of high-order shock-capturing finite difference numerical methods for systems of conservation laws. These methods, called Adaptive Compact Approximation Taylor (ACAT) schemes, use centered $(2p + 1)$-point stencils, where $p$ may take values in $\{1, 2, \dots, P\}$ according to a new family of smoothness indicators in the stencils. The methods are based on a combination of a robust first order scheme and the Compact Approximate Taylor (CAT) methods of order $2p$-order, $p=1,2,\dots, P$ so that they are first order accurate near discontinuities and have order $2p$ in smooth regions, where $(2p +1)$ is the size of the biggest stencil in which large gradients are not detected. CAT methods, introduced in \cite{CP2019}, are an extension to nonlinear problems of the Lax-Wendroff methods in which the Cauchy-Kovalesky (CK) procedure is circumvented following the strategy introduced in \cite{ZBM2017} that allows one to compute time derivatives in a recursive way using high-order centered differentiation formulas combined with Taylor expansions in time. The expression of ACAT methods for 1D and 2D systems of balance laws are given and the performance is tested in a number of test cases for several linear and nonlinear systems of conservation laws, including Euler equations for gas dynamics