Optimal parameters for numerical solvers of PDEs

08/07/2021
by   Gianluca Frasca-Caccia, et al.
0

In this paper we introduce a procedure for identifying optimal methods in parametric families of numerical schemes for initial value problems in partial differential equations. The procedure maximizes accuracy by adaptively computing optimal parameters that minimize a defect-based estimate of the local error at each time-step. Viable refinements are proposed to reduce the computational overheads involved in the solution of the optimization problem, and to maintain conservation properties of the original methods. We apply the new strategy to recently introduced families of conservative schemes for the Korteweg-de Vries equation and for a nonlinear heat equation. Numerical tests demonstrate the improved efficiency of the new technique in comparison with existing methods.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/09/2022

Preconditioned Krylov solvers for structure-preserving discretisations

A key consideration in the development of numerical schemes for time-dep...
research
02/01/2022

Exponentially fitted methods with a local energy conservation law

A new exponentially fitted version of the Discrete Variational Derivativ...
research
06/17/2022

Local Characteristic Decomposition Based Central-Upwind Scheme

We propose novel less diffusive schemes for conservative one- and two-di...
research
05/19/2023

Implicit low-rank Riemannian schemes for the time integration of stiff partial differential equations

We propose two implicit numerical schemes for the low-rank time integrat...
research
06/18/2019

L1-ROC and R2-ROC: L1- and R2-based Reduced Over-Collocation methods for parametrized nonlinear partial differential equations

The onerous task of repeatedly resolving certain parametrized partial di...
research
06/25/2021

Optimal Checkpointing for Adjoint Multistage Time-Stepping Schemes

We consider checkpointing strategies that minimize the number of recompu...
research
10/30/2020

Effects of round-to-nearest and stochastic rounding in the numerical solution of the heat equation in low precision

Motivated by the advent of machine learning, the last few years saw the ...

Please sign up or login with your details

Forgot password? Click here to reset