High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form,typically used in implicit-explicit(IMEX)methods,is not possible.As shown in Boscarino et al.(J.Sci.Comput.68:975-1001,2016)for Runge-Kutta methods,these semi-implicit techniques give a great flexibility,and allow,in many cases,the construction of simple linearly implicit schemes with no need of iterative solvers.In this work,we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype lineal'advection-diffusion equation and in the setting of strong stability preserving(SSP)methods.Our findings are demonstrated on several examples,including nonlinear reaction-diffusion and convection-diffusion problems.

Preface to Focused Section on Efficient High-Order Time Discretization Methods for Partial Differential Equations

During May 8-10,2019,the International Workshop on Efficient High-Order Time Discretization Methods for Partial Differential Equations took place in Villa Orlandi,Anacapri,Italy,a Congress Center of the University of Naples Federico II.About 40 senior researchers,young scholars,and Ph.D.students attended this workshop.The purpose of this event was to explore recent trends and directions in the area of time discretization for the numerical solution of evolutionary partial differential equations with particular application to high-order methods for hyperbolic systems with source and advection-diffusion-reaction equations,and with special emphasis on efficient time-stepping methods such as implicit-explicit(IMEX),semi-implicit and strong stability preserving(SSP)time discretization.The present focused section entitled“Efficient High-Order Time Discretization Methods for Partial Differential Equations”in Communications on Applied Mathematics and Computation(CAMC)consists of five regularly reviewed manuscripts,which were selected from submissions of works presented during the workshop.We thank all the authors of these contributions,and hope that the readers are interested in the topics,techniques and methods,and results of these papers.We also want to thank the CAMC journal editorial staff as well as all the referees for their contributions during the review and publication processes of this focused section.

Implicit-Explicit Runge-Kutta Schemes for the Boltzmann-Poisson System for Semiconductors

In this paper we develop a class of Implicit-Explicit Runge-Kutta schemes for solving the multi-scale semiconductor Boltzmann equation.The relevant scale which characterizes this kind of problems is the diffusive scaling.This means that,in the limit of zero mean free path,the system is governed by a drift-diffusion equation.Our aim is to develop a method which accurately works for the different regimes encountered in general semiconductor simulations:the kinetic,the intermediate and the diffusive one.Moreover,we want to overcome the restrictive time step conditions of standard time integration techniques when applied to the solution of this kind of phenomena without any deterioration in the accuracy.As a result,we obtain high order time and space discretization schemes which do not suffer from the usual parabolic stiffness in the diffusive limit.We show different numerical results which permit to appreciate the performances of the proposed schemes.

A Relaxation Scheme for Solving the Boltzmann Equation Based on the Chapman-Enskog Expansion

In [16] a visco-elastic relaxation system, called the relaxed Burnett system, was proposed by Jin and Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weakly parabolic, has a linearly hyperbolic convection part, and is endowed with a generalized entropy inequality. It agrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman-Enskog expansion. We develop a one-dimensional non-oscillatory numerical scheme based on the relaxed Burnett system for the Boltzmann equation. We compare numerical results for stationary shocks based on this relaxation scheme, and those obtained by the DSMC (Direct Simulation Monte Carlo), by the Navier-Stokes equations and by the extended thermodynamics with thirteen moments (the Grad equations). Our numerical experiments show that the relaxed Burnett gives more accurate approximations to the shock profiles of the Boltzmann equation obtained by the DSMC, for a range of Mach numbers for hypersonic flows, than those obtained by the other hydrodynamic systems.