High-Order Schemes of Exponential Time Differencing for Stiff Systems with Nondiagonal Linear Part
arXiv:2208.14292 · doi:10.1016/j.jcp.2024.113493
Abstract
Exponential time differencing methods is a power tool for high-performance numerical simulation of computationally challenging problems in condensed matter physics, fluid dynamics, chemical and biological physics, where mathematical models often possess fast oscillating or decaying modes -- in other words, are stiff systems. Practical implementation of these methods for the systems with nondiagonal linear part of equations is exacerbated by infeasibility of an analytical calculation of the exponential of a nondiagonal linear operator; in this case, the coefficients of the exponential time differencing scheme cannot be calculated analytically. We suggest an approach, where these coefficients are numerically calculated with auxiliary problems. We rewrite the high-order Runge--Kutta type schemes in terms of the solutions to these auxiliary problems and practically examine the accuracy and computational performance of these methods for a heterogeneous Cahn--Hilliard equation, a sixth-order spatial derivative equation governing pattern formation in the presence of an additional conservation law, and a Fokker--Planck equation governing macroscopic dynamics of a network of neurons.
21 pages, 11 figures
References in corpus (15)
- Destruction of Anderson localization by a weak nonlinearity
- Macroscopic description for networks of spiking neurons
- Pattern formation with a conservation law
- Dynamical mean-field theory and weakly non-linear analysis for the phase separation of active Brownian particles
- A reduction methodology for fluctuation driven population dynamics
- Coherent oscillations in balanced neural networks driven by endogenous fluctuations
- Exact finite-dimensional description for networks of globally coupled spiking neurons
- Unbalanced clustering and solitary states in coupled excitable systems
- Turing Instability in an Economic-Demographic Dynamical System Can Lead to Pattern Formation on Geographical Scale
- Macroscopic behavior of populations of quadratic integrate-and-fire neurons subject to non-Gaussian white noise
- Localization and advectional spreading of convective currents under parametric disorder
- Large-Scale Thermal Convection in a Horizontal Porous Layer
- Diffusion of a passive scalar by convective flows under parametric disorder
- Two scenarios of advective washing-out of localized convective patterns under frozen parametric disorder
- Advectional enhancement of eddy diffusivity under parametric disorder