An a posteriori strategy for adaptive schemes in time and space

TL;DR

This paper introduces an a posteriori adaptive scheme for time and space discretization in convection-diffusion problems, optimizing accuracy and stability by dynamically tuning schemes based on local solution data.

Contribution

It presents a nonlinear adaptive method that adjusts discretization schemes in time and space based on physical data and local regularity, improving efficiency and stability.

Findings

Achieves optimal time-step selection for accurate solutions

Reduces non-physical oscillations in numerical approximations

Demonstrates effectiveness on convection-diffusion problems

Abstract

A nonlinear adaptive procedure for optimising both the schemes in time and space is proposed in view of increasing the numerical efficiency and reducing the computational time. The method is based on a four-parameter family of schemes we shall tune in function of the physical data (velocity, diffusion), the characteristic size in time and space, and the local regularity of the function leading to a nonlinear procedure. The \textit{a posteriori} strategy we adopt consists in, given the solution at time $t^n$, computing a candidate solution with the highest accurate schemes in time and space for all the nodes. Then, for the nodes that present some instabilities, both the schemes in time and space are modified and adapted in order to preserve the stability with a large time step. The updated solution is computed with node-dependent schemes both in time and space. For the sake of simplicity, only convection-diffusion problems are addressed as a prototype with a two-parameters five-points finite difference method for the spatial discretisation together with an explicit time two-parameters four-stages Runge-Kutta method. We prove that we manage to obtain an optimal time-step algorithm that produces accurate numerical approximations exempt of non-physical oscillations.