A recursive system-free single-step temporal discretization method for finite difference methods

02/28/2021
by   Youngjun Lee, et al.
0

Single-stage or single-step high-order temporal discretizations of partial differential equations (PDEs) have shown great promise in delivering high-order accuracy in time with efficient use of computational resources. There has been much success in developing such methods for finite volume method (FVM) discretizations of PDEs. The Picard Integral formulation (PIF) has recently made such single-stage temporal methods accessible for finite difference method (FDM) discretizations. PIF methods rely on the so-called Lax-Wendroff procedures to tightly couple spatial and temporal derivatives through the governing PDE system to construct high-order Taylor series expansions in time. Going to higher than third order in time requires the calculation of Jacobian-like derivative tensor-vector contractions of an increasingly larger degree, greatly adding to the complexity of such schemes. To that end, we present in this paper a method for calculating these tensor contractions through a recursive application of a discrete Jacobian operator that readily and efficiently computes the needed contractions entirely agnostic of the system of partial differential equations (PDEs) being solved.

READ FULL TEXT

Authors

page 12

page 13

page 16

05/29/2020

A single-step third-order temporal discretization with Jacobian-free and Hessian-free formulations for finite difference methods

Discrete updates of numerical partial differential equations (PDEs) rely...
03/16/2021

A Local Deep Learning Method for Solving High Order Partial Differential Equations

At present, deep learning based methods are being employed to resolve th...
07/16/2020

Kronecker Attention Networks

Attention operators have been applied on both 1-D data like texts and hi...
05/06/2022

On the order of accuracy for finite difference approximations of partial differential equations using stencil composition

Stencil composition uses the idea of function composition, wherein two s...
04/05/2019

Tensor B-Spline Numerical Methods for PDEs: a High-Performance Alternative to FEM

Tensor B-spline methods are a high-performance alternative to solve part...
06/21/2018

Optimising finite-difference methods for PDEs through parameterised time-tiling in Devito

Finite-difference methods are widely used in solving partial differentia...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.