Preface

During April 20-22,2022,colleagues and friends gathered at the Institute of Pure&Applied Mathematics(IPAM),at the University of California at Los Angeles to celebrate Professor Stanley Osher's 8Oth birthday in a conference focusing on recent developments in"Optimization,Shape analysis,High-dimensional differential equations in science and Engineering,and machine learning Research(OSHER)"This conference hosted in-person talks by mathematicians,scientists,and industrial professionals worldwide.Those who could not attend extended their warm regards and expressed their appreciation for Professor Osher.

Verification and Validation of High-Resolution Inviscid and Viscous Conical Nozzle Flows

Capturing elaborated flow structures and phenomena is required for well-solved numerical flows.The finite difference methods allow simple discretization of mesh and model equations.However,they need simpler meshes,e.g.,rectangular.The inverse Lax-Wendroff(ILW)procedure can handle complex geometries for rectangular meshes.High-resolution and high-order methods can capture elaborated flow structures and phenomena.They also have strong mathematical and physical backgrounds,such as positivity-preserving,jump conditions,and wave propagation concepts.We perceive an effort toward direct numerical simulation,for instance,regarding weighted essentially non-oscillatory(WENO)schemes.Thus,we propose to solve a challenging engineering application without turbulence models.We aim to verify and validate recent high-resolution and high-order methods.To check the solver accuracy,we solved vortex and Couette flows.Then,we solved inviscid and viscous nozzle flows for a conical profile.We employed the finite difference method,positivity-preserving Lax-Friedrichs splitting,high-resolution viscous terms discretization,fifth-order multi-resolution WENO,ILW,and third-order strong stability preserving Runge-Kutta.We showed the solver is high-order and captured elaborated flow structures and phenomena.One can see oblique shocks in both nozzle flows.In the viscous flow,we also captured a free-shock separation,recirculation,entrainment region,Mach disk,and the diamond-shaped pattern of nozzle flows.

RKDG Methods with Multi-resolution WENO Limiters for Solving Steady-State Problems on Triangular Meshes

In this paper, we design high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (multi-resolution WENO) limiters to compute compressible steady-state problems on triangular meshes. A troubled cell indicator extended from structured meshes to unstructured meshes is constructed to identify triangular cells in which the application of the limiting procedures is required. In such troubled cells, the multi-resolution WENO limiting methods are used to the hierarchical L^(2) projection polynomial sequence of the DG solution. Through using the RKDG methods with multi-resolution WENO limiters, the optimal high-order accuracy can be gradually reduced to first-order in the triangular troubled cells, so that the shock wave oscillations can be well suppressed. In steady-state simulations on triangular meshes, the numerical residual converges to near machine zero. The proposed spatial reconstruction methods enhance the robustness of classical DG methods on triangular meshes. The good results of these RKDG methods with multi-resolution WENO limiters are verified by a series of two-dimensional steady-state problems.

Efficient Finite Difference WENO Scheme for Hyperbolic Systems withNon-conservativeProducts

Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.

Preface

This focused issue of the Communications on Applied Mathematics and Computation is dedicated to the memory of Professor Ching-Shan Chou,who passed away in November 2021.With her passing,our community of applied mathematicians lost not only a brilliant researcher but also a cherished friend and colleague.

A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws

Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations(PDEs).The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions.A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems.Hence,they are easy to be applied to a general hyperbolic system.To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains,inverse Lax-Wendroff(ILW)procedures were developed as a very effective approach in the literature.In this paper,we combine a fifthorder fixed-point fast sweeping WENO method with an ILW procedure to solve steadystate solution of hyperbolic conservation laws on complex computing regions.Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids.Numerical results show highorder accuracy and good performance of the method.Furthermore,the method is compared with the popular third-order total variation diminishing Runge-Kutta(TVD-RK3)time-marching method for steady-state computations.Numerical examples show that for most of examples,the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.

Preface to the Focused Issue in Honor of Professor Tong Zhang on the Occasion of His 90th Birthday

December 9,2022 is the 90th birthday of Tong Zhang,a mathematician in Institute of Mathematics,Chinese Academy of Sciences where he was always working on the Riemann problem for gas dynamics in his mathematical life.To celebrate his 90th birthday and great contributions to this specifc feld,we organize this focused issue in the journal Communications on Applied Mathematics and Computation,since the Riemann problem has been proven to be a building block in all felds of theory,numerics and applications of hyperbolic conservation laws.

Preface to the Focused Issue on WENO Schemes

The weighted essentially-oscillatory(WENO)schemes are a class of finite volume and finite difference methods for solving convection-dominated problems,mainly hyperbolic conservation laws.The idea comes from the earlier essentially-oscillatory(ENO)schemes,first developed in[1]in finite volume version and in[6]in finite difference version,for solving hyperbolic conservation laws.

激波与旋涡对干扰研究：激波动力学特性及声波产生机理

采用五阶WENO格式，通过数值求解二维非定常Navier-Stokes方程，系统模拟了激波与旋涡对的干扰过程。我们研究了强度为1．05和1．2两个典型的激波同不同强度的旋涡对的干扰过程。根据激波动力学特性，我们把激波与掠过型旋涡对干扰分成四类，而碰撞型干扰分成七种不同的类型。研究表明，激波与旋涡对干扰过程中，声波产生有三种机制，激波与涡对的干扰，涡对之间的耦合以及激波与声波的干扰。

A REACTIVE DYNAMIC CONTINUUM USER EQUILIBRIUM MODEL FOR BI-DIRECTIONAL PEDESTRIAN FLOWS

In this paper, a reactive dynamic user equilibrium model is extended to simulate two groups of pedestrians traveling on crossing paths in a continuous walking facility. Each group makes path choices to minimize the travel cost to its destination in a reactive manner based on instantaneous information. The model consists of a conservation law equation coupled with an Eikonal-type equation for each group. The velocity-density relationship of pedestrian movement is obtained via an experimental method. The model is solved using a finite volume method for the conservation law equation and a fast-marching method for the Eikonal-type equation on unstructured grids. The numerical results verify the rationality of the model and the validity of the numerical method. Based on this continuum model, a number of results, e.g., the formation of strips or moving clusters composed of pedestrians walking to the same destination, are also observed.

Numerical simulations of compressible mixing layers with a discontinuous Galerkin method

Discontinuous Galerkin（DG） method is known to have several advantages for flow simulations,in particular,in fiexible accuracy management and adaptability to mesh refinement. In the present work,the DG method is developed for numerical simulations of both temporally and spatially developing mixing layers. For the temporally developing mixing layer,both the instantaneous fiow field and time evolution of momentum thickness agree very well with the previous results. Shocklets are observed at higher convective Mach numbers and the vortex paring manner is changed for high compressibility. For the spatially developing mixing layer,large-scale coherent structures and self-similar behavior for mean profiles are investigated. The instantaneous fiow field for a three-dimensional compressible mixing layer is also reported,which shows the development of largescale coherent structures in the streamwise direction. All numerical results suggest that the DG method is effective in performing accurate numerical simulations for compressible shear fiows.

Convergence to Steady-State Solutions of the New Type of High-Order Multi-resolution WENO Schemes: a Numerical Study

A new type of high-order multi-resolution weighted essentially non-oscillatory(WENO)schemes(Zhu and Shu in J Comput Phys,375:659-683,2018)is applied to solve for steady-state problems on structured meshes.Since the classical WENO schemes(Jiang and Shu in J Comput Phys,126:202-228,1996)might suffer from slight post-shock oscillations(which are responsible for the residue to hang at a truncation error level),this new type of high-order finite-difference and finite-volume multi-resolution WENO schemes is applied to control the slight post-shock oscillations and push the residue to settle down to machine zero in steady-state simulations.This new type of multi-resolution WENO schemes uses the same large stencils as that of the same order classical WENO schemes,could obtain fifth-order,seventh-order,and ninth-order in smooth regions,and could gradually degrade to first-order so as to suppress spurious oscillations near strong discontinuities.The linear weights of such new multi-resolution WENO schemes can be any positive numbers on the condition that their sum is one.This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finitedifference and finite-volume WENO schemes for solving steady-state problems.In comparison with the classical fifth-order finite-difference and finite-volume WENO schemes,the residue of these new high-order multi-resolution WENO schemes can converge to a tiny number close to machine zero for some benchmark steady-state problems.

Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations

In this paper,we study the superconvergence properties of the energy-conserving discontinuous Galerkin(DG)method in[18]for one-dimensional linear hyperbolic equations.We prove the approximate solution superconverges to a particular projection of the exact solution.The order of this superconvergence is proved to be k+2 when piecewise Pk polynomials with K≥1 are used.The proof is valid for arbitrary non-uniform regular meshes and for piecewise polynomials with arbitrary K≥1.Furthermore,we find that the derivative and function value approxi?mations of the DG solution are superconvergent at a class of special points,with an order of k+1 and R+2,respectively.We also prove,under suitable choice of initial discretization,a(2k+l)-th order superconvergence rate of the DG solution for the numerical fluxes and the cell averages.Numerical experiments are given to demonstrate these theoretical results.

高精度WENO格式的发展与展望

加权本质无振荡(weighted essentially non-oscillatory, WENO)格式是用于求解双曲守恒律方程和对流占优问题的一类高精度数值方法. WENO格式设计的思想是在解的光滑区域中获得高阶数值精度,而在解的间断附近保持本质无振荡的性质.以这种思想设计的有限差分和有限体积高精度WENO格式在计算流体力学等领域中得到了广泛应用.本文首先回顾WENO格式设计的基本思想和性质,简要介绍近年来WENO格式研究方面的一些进展,并阐述US-WENO (unequal-sized WENO)格式、MR-WENO (multi-resolution WENO)格式和HWENO (Hermite WENO)格式的构造策略.此外,本文还介绍高精度WENO格式在结构网格和非结构网格上的一些进展,展望这些高精度格式在多个领域中的应用以及未来的发展趋势.

Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity

A time discretization method is called strongly stable(or monotone),if the norm of its numerical solution is nonincreasing.Although this property is desirable in various of contexts,many explicit Runge-Kutta(RK)methods may fail to preserve it.In this paper,we enforce strong stability by modifying the method with superviscosity,which is a numerical technique commonly used in spectral methods.Our main focus is on strong stability under the inner-product norm for linear problems with possibly non-normal operators.We propose two approaches for stabilization:the modified method and the filtering method.The modified method is achieved by modifying the semi-negative operator with a high order superviscosity term;the filtering method is to post-process the solution by solving a diffusive or dispersive problem with small superviscosity.For linear problems,most explicit RK methods can be stabilized with either approach without accuracy degeneration.Furthermore,we prove a sharp bound(up to an equal sign)on diffusive superviscosity for ensuring strong stability.For nonlinear problems,a filtering method is investigated.Numerical examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance.

Preface to Focused Issue on Discontinuous Galerkin Methods

The discontinuous Galerkin(DG)method is a class of finite element methods using com-pletely discontinuous piecewise smooth functions(typically polynomials)as basis and test functions.Since its inception in 1973[10],it has seen a sustained development,both in the computational mathematics community and in many scientific and engineering application communities.The DG methods have several advantages,such as its extreme flexibility in dealing with complex geometry and adaptive computation(both h-and p-adaptivities are easy to implement),extremely high parallel efficiency,good stability properties(energy and entropy stability has been established for DG methods in many situations),nice con-vergence and superconvergence properties,and capability to solve hyperbolic and convec-tion-dominated problems effectively.

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.

Preface to the Focused Issue in Honor of Professor Philip Roe on the Occasion of His 80th Birthday

On July 3-4,2018,an international workshop was organised at the University of Cambridge,by Dr.Nikoforakis,to mark the 80th anniversary of a young scientist,Philip Roe.Phil's scientific achievements are numerous,and it is quite difficult to adequately address his accomplishments in a short preface.Phil began his journey at Cambridge,first for his Bachelor's degree and then obtaining his Master's degree in 1962.He left for the Royal Aircraft Establishment(RAE)between 1962 to 1984 to work as an engineer.It was a period of adventure,and the design of mature CFD codes for solving the Euler equations was far from being achieved.Though the Lax-Wendroff theorem was known(published in 1960),as for many novelties,it had taken some time before it became a gold standard.At the RAE,Phil was assigned by his manager to work in a direction that eventually led to Roe's Riemann solver and flux.This method is now standard in every specialised textbook,but the jump from generalizing the Murman scheme for Euler equations to what is now commonly known as the Roe flux is far from obvious.Phil Roe relates this adventure,and others,in his"scientific autobiography",one of the papers of this focused issue.

Preface to the Focused Issue on Fractional Derivatives and General Nonlocal Models

Fractional calculus,which has two prominent characteristics-singularity and nonlocality,comprises the integration and differentiation of any positive real(and even complex)order.It has a more than three-hundred-year history and can be traced back to a letter from Leib-niz to UHopital,dated 30 September 1695,in which the meaning of the one-half order derivative was first discussed and some remarks about its feasibility were made.Abel was probably the first who rendered an application of fractional calculus.He used the derivatives of arbitrary order to solve the tautochrone(isochrone)problem in 1823.Fractional calculus underwent two periods:from its beginning to the 1970s,and after 1970s.During the first period,fractional calculus was studied mainly by mathematicians as an abstract field containing only pure mathematical manipulations of little applications,except for sporadic applications in rheology.During the second period,the paradigm shifted from pure mathematical research to applications in various realms,including anomalous diffusion,anomalous convection,power laws,allometric scaling laws,history dependence,long?range interactions,and so on.

Preface

This is a focused issue dedicated to the memory of the late Professor Ben-yu Guo(1942-2016),a prominent numerical analyst at Shanghai University and Shanghai Normal University,and a prolific researcher with more than 300 peer-reviewed publications,many of which are in prestigious journals.His work has been well recognized in the world and extensively cited.He received numerous prestigious awards,including a degree of Doctor of Science honoris causa from Sanford University in England.