Dimension by Dimension Finite Volume HWENO Method for Hyperbolic Conservation Laws

In this paper,we propose a finite volume Hermite weighted essentially non-oscillatory(HWENO)method based on the dimension by dimension framework to solve hyperbolic conservation laws.It can maintain the high accuracy in the smooth region and obtain the high resolution solution when the discontinuity appears,and it is compact which will be good for giving the numerical boundary conditions.Furthermore,it avoids complicated least square procedure when we implement the genuine two dimensional(2D)finite volume HWENO reconstruction,and it can be regarded as a generalization of the one dimensional(1D)HWENO method.Extensive numerical tests are performed to verify the high resolution and high accuracy of the scheme.

On High-Resolution Entropy-Consistent Flux with Slope Limiter for Hyperbolic Conservation Laws

This paper proposes a new version of the high-resolution entropy-consistent(EC-Limited)flux for hyperbolic conservation laws based on a new minmod-type slope limiter.Firstly,we identify the numerical entropy production,a third-order differential term deduced from the previous work of Ismail and Roe[11].The corresponding dissipation term is added to the original Roe flux to achieve entropy consistency.The new,resultant entropy-consistent(EC)flux has a general and explicit analytical form without any corrective factor,making it easy to compute and a less-expensive method.The inequality constraints are imposed on the standard piece-wise quadratic reconstruction to enforce the pointwise values of bounded-type numerical solutions.We design the new minmod slope limiter as combining two separate limiters for left and right states.We propose the EC-Limited flux by adding this reconstruction data method to the primitive variables rather than to the conservative variables of the EC flux to preserve the equilibrium of the primitive variables.These resulting fluxes are easily applied to general hyperbolic conservation laws while having attractive features:entropy-stable,robust,and non-oscillatory.To illustrate the potential of these proposed fluxes,we show the applications to the Burgers equation and the Euler equations.

A local pseudo arc-length method for hyperbolic conservation laws

A local pseudo arc-length method（LPALM）for solving hyperbolic conservation laws is presented in this paper.The key idea of this method comes from the original arc-length method,through which the critical points are bypassed by transforming the computational space.The method is based on local changes of physical variables to choose the discontinuous stencil and introduce the pseudo arc-length parameter,and then transform the governing equations from physical space to arc-length space.In order to solve these equations in arc-length coordinate,it is necessary to combine the velocity of mesh points in the moving mesh method,and then convert the physical variable in arclength space back to physical space.Numerical examples have proved the effectiveness and generality of the new approach for linear equation,nonlinear equation and system of equations with discontinuous initial values.Non-oscillation solution can be obtained by adjusting the parameter and the mesh refinement number for problems containing both shock and rarefaction waves.

High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

In this paper,the maximum-principle-preserving(MPP)and positivitypreserving(PP)flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes(WCNSs)for scalar conservation laws and the compressible Euler systems in both one and two dimensions.The main idea of the present method is to rewrite the scheme in a conservative form,and then define the local limiting parameters via case-by-case discussion.Smooth test problems are presented to demonstrate that the proposed MPP/PP WCNSs incorporating a third-order Runge-Kutta method can attain the desired order of accuracy.Other test problems with strong shocks and high pressure and density ratios are also conducted to testify the performance of the schemes.

SPECTRAL/HP ELEMENT METHOD WITH HIERARCHICAL RECONSTRUCTION FOR SOLVING NONLINEAR HYPERBOLIC CONSERVATION LAWS

The hierarchical reconstruction （HR） [Liu, Shu, Tadmor and Zhang, SINUM ＇07] has been successfully applied to prevent oscillations in solutions computed by finite volume, Runge-Kutta discontinuous Galerkin, spectral volume schemes for solving hyperbolic conservation laws. In this paper, we demonstrate that HR can also be combined with spectral/hp element method for solving hyperbolic conservation laws. An orthogonal spectral basis written in terms of Jacobi polynomials is applied. High computational efficiency is obtained due to such matrix-free algorithm. The formulation is conservative, and essential nomoscillation is enforced by the HR limiter. We show that HR preserves the order of accuracy of the spectral/hp element method for smooth solution problems and generate essentially non-oscillatory solutions profiles for capturing discontinuous solutions without local characteristic decomposition. In addition, we introduce a postprocessing technique to improve HR for limiting high degree numerical solutions.

MmB DIFFERENCE SCHEMES FOR TWODIMENSIONAL HYPERBOLIC CONSERVATION LAWS

A new class of second order accuracy semidiscrete difference schemes is presented for the two-dimensional nonlinear scalar hyperbolic conservation laws. It is based on flux splitting, piecewise linear cell-averaged reconstruction and upwind property in the spatial discretization. By using TVD Runge-Kutta time discretization method, the full discrete scheme is obtained and its MmB property is proved. The extension to the two-dimensionalnonlinear hyperbolic conservation law systems is straightforward by using component-wise manner. The main advantage is simple: no Riemann problem is solved, and so field-by-field decomposition is avoided and the complicated computation is reduced. Numerical results of two-dimensional Euler equations of compressible gas dynamics verify the accuracy and robustness of the method.

LARGE TIME STEP GENERALIZATION OF RANDOM CHOICE FINITE DIFFERENCE SCHEME FOR HYPERBOLIC CONSERVATION LAWS

A natural generalization of random choice finite difference scheme of Harten and Lax for Courant number larger than 1 is obtained. We handle interactions between neighboring Riemann solvers by linear superposition of their conserved quantities. We show consistency of the scheme for arbitrarily large Courant numbers. For scalar problems the scheme is total variation diminishing.A brief discussion is given for entropy condition.

A NOTE ON THE RIEMANN PROBLEM TO HYPERBOLIC CONSERVATION LAWS

This paper is a extension of [1], [3]. By the method in [1], the authors prove the global existence of the solutions of the Riemann problem for a class of 2 × 2 nonstrictly hyperbolic conservation laws.

An oscillation-free Hermite WENO scheme for hyperbolic conservation laws

In this paper, the sixth-order oscillation-free Hermite weighted essentially non-oscillatory (OFHWENO) scheme is proposed for hyperbolic conservation laws on structured meshes, where the zeroth- andfirst-order moments are the variables for the governing equations. The main difference from other HWENOschemes existing in the literature is that we add high-order numerical damping terms in the first-order momentequations to control spurious oscillations for the OF-HWENO scheme. The OF-HWENO scheme not only canachieve the designed optimal numerical order, but also can be easily implemented as we use only one set ofstencils in the reconstruction procedure and the same reconstructed polynomials are applied for the zeroth- andfirst-order moment equations. In order to obtain the adaptive order resolution when facing discontinuities, atransition polynomial is added in the reconstruction, where the associated linear weights can also be any positivenumbers as long as their summation equals one. In addition, the OF-HWENO scheme still keeps compactnessas only immediate neighbor values are needed in the space discretization. Some benchmark numerical tests areperformed to illustrate the high-order accuracy, high resolution and robustness of the proposed scheme.

The Formulation of Finite Difference RBFWENO Schemes for Hyperbolic Conservation Laws:An Alternative Technique

To solve conservation laws,efficient schemes such as essentially nonoscillatory(ENO)and weighted ENO(WENO)have been introduced to control the Gibbs oscillations.Based on radial basis functions(RBFs)with the classical WENO-JS weights,a new type of WENO schemes has been proposed to solve conservation laws[J.Guo et al.,J.Sci.Comput.,70(2017),pp.551–575].The purpose of this paper is to introduce a new formulation of conservative finite difference RBFWENO schemes to solve conservation laws.Unlike the usual method for reconstructing the flux functions,the flux function is generated directly with the conservative variables.Comparing with Guo and Jung(2017),the main advantage of this framework is that arbitrary monotone fluxes can be employed,while in Guo and Jung(2017)only smooth flux splitting can be used to reconstruct flux functions.Several 1D and 2D benchmark problems are prepared to demonstrate the good performance of the new scheme.

Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws

This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws.The basic idea is that the“meaningful objects”are the fluxes,evaluated across domain boundaries over time intervals.The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting.It implies that a weak solution indeed satisfies the balance law.In fact,it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary.It should be emphasized that the weak solutions considered here need not be entropy solutions.Furthermore,the assumption imposed on the flux f(u)is quite minimal-just that it is locally bounded.

Moment-Based Multi-Resolution HWENO Scheme for Hyperbolic Conservation Laws

In this paper,a high-order moment-based multi-resolution Hermite weighted essentially non-oscillatory(HWENO)scheme is designed for hyperbolic conservation laws.The main idea of this scheme is derived from our previous work[J.Comput.Phys.,446(2021)110653],in which the integral averages of the function and its first order derivative are used to reconstruct both the function and its first order derivative values at the boundaries.However,in this paper,only the function values at the Gauss-Lobatto points in the one or two dimensional case need to be reconstructed by using the information of the zeroth and first order moments.In addition,an extra modification procedure is used to modify those first order moments in the troubledcells,which leads to an improvement of stability and an enhancement of resolution near discontinuities.To obtain the same order of accuracy,the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as the general HWENO scheme and is more compact than the generalWENO scheme.Moreover,the linear weights are not unique and are independent of the node position,and the CFL number can still be 0.6whether for the one or two dimensional case,which has to be 0.2 in the two dimensional case for other HWENO schemes.Extensive numerical examples are given to demonstrate the stability and resolution of such moment-based multi-resolution HWENO scheme.

ON THE CENTRAL RELAXING SCHEME Ⅱ: SYSTEMS OF HYPERBOLIC CONSERVATION LAWS

This paper continues to study the central relaxing schemes for system of hyperbolic conservation laws, based on the local relaxation approximation. Two classes of relaxing systems with stiff source term are introduced to approximate system of conservation laws in curvilinear coordinates. Based on them, the semi-implicit relaxing schemes are con- structed as in [6, 12] without using any linear or nonlinear Riemann solvers. Numerical experiments for one-dimensional and two-dimensional problems are presented to demon- strate the performance and resolution of the current schemes.

Improved Symmetry Property of High Order Weighted Essentially Non-Oscillatory Finite Difference Schemes for Hyperbolic Conservation Laws

This study aims to investigate the rapid loss of numerical symmetry for problems with symmetrical initial conditions and boundary conditions when solved by the seventh and higher order nonlinear characteristic-wise weighted essentially non-oscillatory(WENO)finite difference schemes.Using the one-dimensional double rarefaction wave problem and the Sedov blast-wave problems,and the twodimensional Rayleigh-Taylor instability(RTI)problem as examples,we illustrate numerically that the sensitive interaction of the round-off error due to the numerical unstable explicit form of the local lower order smoothness indicators in the nonlinear weights definition,which are often given and used in the literature,and the nonlinearity of the WENO scheme are responsible for the rapid growth of asymmetry of an otherwise symmetric problem.An equivalent but compact and numerical stable compact form of the local lower order smoothness indicators is suggested for delaying the onset of and reducing the magnitude of the symmetry error.The benefits of using the compact form of the local lower order smoothness indicators should also be applicable to non-symmetrical strongly non-linear problems in terms of improved numerical stability,reduced rounding errors and increased computational efficiency.

A Cartesian Embedded Boundary Method for Hyperbolic Conservation Laws

We develop an embedded boundary finite difference technique for solving the compressible two-or three-dimensional Euler equations in complex geometries on a Cartesian grid.The method is second order accurate with an explicit time step determined by the grid size away from the boundary.Slope limiters are used on the embedded boundary to avoid non-physical oscillations near shock waves.We show computed examples of supersonic flow past a cylinder and compare with results computed on a body fitted grid.Furthermore,we discuss the implementation of the method for thin geometries,and show computed examples of transonic flow past an airfoil.

Hyperbolic Conservation Laws on Manifolds.An Error Estimate for Finite Volume Schemes

Following Ben-Artzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1-error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L1 norm is of order h1/4 at most, where h represents the maximal diameter of elements in the family of geodesic triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and LeFloch＇s theory which was originally developed in the Euclidian setting. We extend the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties.

Multiple Shock Fronts for Hyperbolic Conservation Laws in Higher Dimensional Space

The local existence of multiple shock fronts for hyperbolic conservation laws in higher dimensional space is established under the assumption that its frozen problem produces multiple uniformly stable planar shock fronts.

Boundary Layer to a System of Viscous Hyperbolic Conservation Laws

In this paper, we investigate the large-time behavior of solutions to the initial-boundary value problem for n × n hyperbolic system of conservation laws with artificial viscosity in the half line （0, ∞）. We first show that a boundary layer exists if the corresponding hyperbolic part contains at least one characteristic field with negative propagation speed. We further show that such boundary layer is nonlinearly stable under small initial perturbation. The proofs are given by an elementary energy method.

Numerical Issues in the Implementation of High Order Polynomial Multi-Domain Penalty Spectral Galerkin Methods for Hyperbolic Conservation Laws

In this paper,we consider high order multi-domain penalty spectral Galerkin methods for the approximation of hyperbolic conservation laws.This formulation has a penalty parameter which can vary in space and time,allowing for flexibility in the penalty formulation.This flexibility is particularly advantageous for problems with an inhomogeneous mesh.We show that the discontinuous Galerkin method is equivalent to the multi-domain spectral penalty Galerkin method with a particular value of the penalty parameter.The penalty parameter has an effect on both the accuracy and stability of the method.We examine the numerical issues which arise in the implementation of high order multi-domain penalty spectral Galerkin methods.The coefficient truncation method is proposed to prevent the rapid error growth due to round-off errors when high order polynomials are used.Finally,we show that an inconsistent evaluation of the integrals in the penalty method may lead to growth of errors.Numerical examples for linear and nonlinear problems are presented.

AN EFFICIENT THIRD-ORDER SCHEME FOR THREE-DIMENSIONAL HYPERBOLIC CONSERVATION LAWS

In this paper,we present a third-order central weighted essentially nonoscillatory(CWENO)reconstruction for computations of hyperbolic conservation laws in three space dimensions.Simultaneously,as a Godunov-type central scheme,the CWENOtype central-upwind scheme,i.e.,the semi-discrete central-upwind scheme based on our third-order CWENO reconstruction,is developed straightforwardly to solve 3D systems by the so-called componentwise and dimensional-by-dimensional technologies.The high resolution,the efficiency and the nonoscillatory property of the scheme can be verified by solving several numerical experiments.