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.

High Order Conservative Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

In this paper,we present a conservative semi-Lagrangian finite-difference scheme for the BGK model.Classical semi-Lagrangian finite difference schemes,coupled with an L-stable treatment of the collision term,allow large time steps,for all the range of Knudsen number[17,27,30].Unfortunately,however,such schemes are not conservative.Lack of conservation is analyzed in detail,and two main sources are identified as its cause.First,when using classical continuous Maxwellian,conservation error is negligible only if velocity space is resolved with sufficiently large number of grid points.However,for a small number of grid points in velocity space such error is not negligible,because the parameters of the Maxwellian do not coincide with the discrete moments.Secondly,the non-linear reconstruction used to prevent oscillations destroys the translation invariance which is at the basis of the conservation properties of the scheme.As a consequence the schemes show a wrong shock speed in the limit of small Knudsen number.To treat the first problem and ensure machine precision conservation of mass,momentum and energy with a relatively small number of velocity grid points,we replace the continuous Maxwellian with the discrete Maxwellian introduced in[22].The second problem is treated by implementing a conservative correction procedure based on the flux difference form as in[26].In this way we can construct conservative semi-Lagrangian schemes which are Asymptotic Preserving(AP)for the underlying Euler limit,as the Knudsen number vanishes.The effectiveness of the proposed scheme is demonstrated by extensive numerical tests.