On different flux splittings and flux functions in WENO schemes for balance laws

In this paper we focus our attention on obtaining well-balanced schemes for balance laws by using Marquina’s flux in combination with the finite difference and finite volume WENO schemes. We consider also the Rusanov flux splitting and the HLL approximate Riemann solver. In particular, for the presented numerical schemes we develop corresponding discretizations of the source term, based on the idea of balancing with the flux gradient. When applied to the open-channel flow and to the shallow water equations, we obtain the finite difference WENO scheme with Marquina’s flux splitting, which satisfies the approximate conservation property, and also the balanced finite volume WENO scheme with Marquina’s solver satisfying the exact conservation property. Finally, we also present an improvement of the balanced finite difference WENO scheme with the Rusanov (locally Lax–Friedrichs) flux splitting, we previously developed in [Vuković S, Sopta L. ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. J Comput Phys 2002;179:593–621].     

Upwind methods for hyperbolic conservation laws with source terms

This paper deals with the extension of some upwind schemes to hyperbolic systems of conservation laws with source term. More precisely we give methods to get natural upwind discretizations of the source term when the flux is approximated by using flux-difference or flux-splitting techniques. In particular, the Q-schemes of Roe and van Leer and the flux-splitting techniques of Steger-Warming and Vijayasundaram are considered. Numerical results for a scalar advection equation with nonlinear source and for the one-dimensional shallow water equations are presented. In the last case we compare the different schemes proposed in terms of a conservation property. When this property does not hold, spurious numerical waves can appear which is the case for the centred discretization of the source term.     

Lax–Friedrichs Multigrid Fast Sweeping Methods for Steady State Problems for Hyperbolic Conservation Laws

Fast sweeping methods are efficient Gauss–Seidel iterative numerical schemes originally designed for solving static Hamilton–Jacobi equations. Recently, these methods have been applied to solve hyperbolic conservation laws with source terms. In this paper, we propose Lax–Friedrichs fast sweeping multigrid methods which allow even more efficient calculations of viscosity solutions of stationary hyperbolic problems. Due to the choice of Lax–Friedrichs numerical fluxes, general problems can be solved without difficult inversion. High order discretization, e.g., WENO finite difference method, can be incorporated to achieve high order accuracy. On the other hand, multigrid methods, which have been widely used to solve elliptic equations, can speed up the computation by smoothing errors of low frequencies on coarse meshes. We modify the classical multigrid method with regard to properties of viscous solutions to hyperbolic conservation equations by introducing WENO interpolation between levels of mesh grids. Extensive numerical examples in both scalar and system test problems in one and two dimensions demonstrate the efficiency, high order accuracy and the capability of resolving singularities of the viscosity solutions.     

Computational models for weakly dispersive nonlinear water waves

Numerical methods for the two- and three-dimensional Boussinesq equations governing weakly nonlinear and dispersive water waves are presented and investigated. Convenient handling of grids adapted to the geometry or bottom topography is enabled by finite element discretization in space. Staggered finite difference schemes are used for the temporal discretization, resulting in only two linear systems to be solved during each time step. Efficient iterative solution of linear systems is discussed. By introducing correction terms in the equations, a fourth-order, two-level temporal scheme can be obtained. Combined with (bi-) quadratic finite elements, the truncation errors of this scheme can be made of the same order as the neglected perturbation terms in the analytical model, provided that the element size is of the same order as the characteristic depth. We present analysis of the proposed schemes in terms of numerical dispersion relations. Verification of the schemes and their implementations is performed for standing waves in a closed basin with constant depth. More challenging applications cover plane incoming waves on a curved beach and earthquake induced waves over a shallow seamount. In the latter example we demonstrate a significantly increased computational efficiency when using higher-order schemes and bathymetry-adapted finite element grids.     

Hybrid Well-balanced WENO Schemes with Different Indicators for Shallow Water Equations

In (J. Comput. Phys. 229: 8105–8129, 2010), Li and Qiu investigated the hybrid weighted essentially non-oscillatory (WENO) schemes with different indicators for Euler equations of gas dynamics. In this continuation paper, we extend the method to solve the one- and two-dimensional shallow water equations with source term due to the non-flat bottom topography, with a goal of obtaining the same advantages of the schemes for the Euler equations, such as the saving computational cost, essentially non-oscillatory property for general solution with discontinuities, and the sharp shock transition. Extensive simulations in one- and two-dimensions are provided to illustrate the behavior of this procedure.     

A family of stable numerical solvers for the shallow water equations with source terms

In this work we introduce a multiparametric family of stable and accurate numerical schemes for 1D shallow water equations. These schemes are based upon the splitting of the discretization of the source term into centered and decentered parts. These schemes are specifically designed to fulfill the enhanced consistency condition of Bermúdez and Vázquez, necessary to obtain accurate solutions when source terms arise. Our general family of schemes contains as particular cases the extensions already known of Roe and Van Leer schemes, and as new contributions, extensions of Steger–Warming, Vijayasundaram, Lax–Friedrichs and Lax–Wendroff schemes with and without flux-limiters. We include some meaningful numerical tests, which show the good stability and consistency properties of several of the new methods proposed. We also include a linear stability analysis that sets natural sufficient conditions of stability for our general methods.     

A Positivity-Preserving Well-Balanced Central Discontinuous Galerkin Method for the Nonlinear Shallow Water Equations

In this paper, we consider the development of central discontinuous Galerkin methods for solving the nonlinear shallow water equations over variable bottom topography in one and two dimensions. A reliable numerical scheme for these equations should preserve still-water stationary solutions and maintain the non-negativity of the water depth. We propose a high-order technique which exactly balances the flux gradients and source terms in the still-water stationary case by adding correction terms to the base scheme, meanwhile ensures the non-negativity of the water depth by using special approximations to the bottom together with a positivity-preserving limiter. Numerical tests are presented to illustrate the accuracy and validity of the proposed schemes.     

Hybrid Finite Difference Weighted Essentially Non-oscillatory Schemes for the Compressible Ideal Magnetohydrodynamics Equation

In this paper, we present hybrid weighted essentially non-oscillatory (WENO) schemes with several discontinuity detectors for solving the compressible ideal magnetohydrodynamics (MHD) equation. Li and Qiu (J Comput Phys 229:8105–8129, 2010) examined effectiveness and efficiency of several different troubled-cell indicators in hybrid WENO methods for Euler gasdynamics. Later, Li et al. (J Sci Comput 51:527–559, 2012) extended the hybrid methods for solving the shallow water equations with four better indicators. Hybrid WENO schemes reduce the computational costs, maintain non-oscillatory properties and keep sharp transitions for problems. The numerical results of hybrid WENO-JS/WENO-M schemes are presented to compare the ability of several troubled-cell indicators with a variety of test problems. The focus of this paper, we propose optimal and reliable indicators for performance comparison of hybrid method using troubled-cell indicators for efficient numerical method of ideal MHD equations. We propose a modified ATV indicator that uses a second derivative. It is advantageous for differential discontinuity detection such as jump discontinuity and kink. A detailed numerical study of one-dimensional and two-dimensional cases is conducted to address efficiency (CPU time reduction and more accurate numerical solution) and non-oscillatory property problems. We demonstrate that the hybrid WENO-M scheme preserves the advantages of WENO-M and the ratio of computational costs of hybrid WENO-M and hybrid WENO-JS is smaller than that of WENO-M and WENO-JS.     

Second-order-accurate explicit fluctuation splitting schemes for unsteady problems

This paper is concerned with the issue of obtaining explicit fluctuation splitting schemes which achieve second-order accuracy in both space and time on an arbitrary unstructured triangular mesh. A theoretical analysis demonstrates that, for a linear reconstruction of the solution, mass lumping does not diminish the accuracy of the scheme provided that a Galerkin space discretization is employed. Thus, two explicit fluctuation splitting schemes are devised which are second-order accurate in both space and time, namely, the well known Lax-Wendroff scheme and a Lax-Wendroff-type scheme using a three-point-backward discretization of the time derivative. A thorough mesh-refinement study verifies the theoretical order of accuracy of the two schemes on meshes with increasing levels of nonuniformity.     

High-Order Well-Balanced Finite Difference WENO Schemes for a Class of Hyperbolic Systems with Source Terms

In this paper, we generalize the high order well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, designed earlier by us in Xing and Shu (2005, J. Comput. phys. 208, 206–227) for the shallow water equations, to solve a wider class of hyperbolic systems with separable source terms including the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. Properties of the scheme for the shallow water equations (Xing and Shu 2005, J. Comput. phys. 208, 206–227), such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions, are maintained for the scheme when applied to this general class of hyperbolic systems     

Implicit–Explicit Schemes for BGK Kinetic Equations

In this work a new class of numerical methods for the BGK model of kinetic equations is presented. In principle, schemes of any order of accuracy in both space and time can be constructed with this technique. The methods proposed are based on an explicit–implicit time discretization. In particular the convective terms are treated explicitly, while the source terms are implicit. In this fashion even problems with infinite stiffness can be integrated with relatively large time steps. The conservation properties of the schemes are investigated. Numerical results are shown for schemes of order 1, 2 and 5 in space, and up to third-order accurate in time.     

A Numerical Strategy for Freestream Preservation of the High Order Weighted Essentially Non-oscillatory Schemes on Stationary Curvilinear Grids

The weighted essentially non-oscillatory (WENO) schemes have been extensively employed for the simulation of complex flow fields due to their high order accuracy and good shock-capturing properties. However, the standard finite difference WENO scheme cannot hold freestream automatically in general curvilinear coordinates. Numerical errors from non-preserved freestream can hide small scales such as turbulent flow structures; aero-acoustic waves which can make the results inaccurate or even cause the simulation failure. To address this issue, a new numerical strategy to ensure freestream preservation properties of the WENO schemes on stationary curvilinear grids is proposed in this paper. The essential idea of this approach is to offset the geometrically induced errors by proper discretization of the metric invariants. It includes the following procedures: (1) the metric invariants are retained in the governing equations and the full forms of the transformed equations on the general curvilinear coordinates are solved; (2) the symmetrical, conservative form of the metrics instead of the original ones are used; (3) the WENO schemes which are applied for the inviscid fluxes of the governing equations are employed to compute the outer-level partial derivatives of the metric invariants. In other words, the outer-level derivative operators for the metric invariants are kept the same with those for the corresponding inviscid fluxes. It is verified theoretically in this paper that by using this approach, the WENO schemes hold the freestream preservation properties naturally and thus work well in the generalized coordinate systems. For some well-known WENO schemes, the derivative operators for the metric invariants are explicitly expressed and thus this approach can be straightforwardly employed. The effectiveness of this strategy is validated by several benchmark test cases.     

High Order Well-Balanced WENO Scheme for the Gas Dynamics Equations Under Gravitational Fields

The gas dynamics equations, coupled with a static gravitational field, admit the hydrostatic balance where the flux produced by the pressure is exactly canceled by the gravitational source term. Many astrophysical problems involve the hydrodynamical evolution in a gravitational field, therefore it is essential to correctly capture the effect of gravitational force in the simulations. Improper treatment of the gravitational force can lead to a solution which either oscillates around the equilibrium, or deviates from the equilibrium after a long time run. In this paper we design high order well-balanced finite difference WENO schemes to this system, which can preserve the hydrostatic balance state exactly and at the same time can maintain genuine high order accuracy. Numerical tests are performed to verify high order accuracy, well-balanced property, and good resolution for smooth and discontinuous solutions. The main purpose of the well-balanced schemes designed in this paper is to well resolve small perturbations of the hydrostatic balance state on coarse meshes. The more difficult issue of convergence towards such hydrostatic balance state from an arbitrary initial condition is not addressed in this paper.     

ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows

We develop a new family of well-balanced path-conservative quadrature-free one-step ADER finite volume and discontinuous Galerkin finite element schemes on unstructured meshes for the solution of hyperbolic partial differential equations with non-conservative products and stiff source terms. The fully discrete formulation is derived using the recently developed framework of explicit one-step P_NP_M schemes of arbitrary high order of accuracy in space and time for conservative hyperbolic systems [Dumbser M, Balsara D, Toro EF, Munz CD. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J Comput Phys 2008;227:8209–53]. The two key ingredients of our high order approach are: first, the high order accurate P_NP_M reconstruction operator on unstructured meshes, using the WENO strategy presented in [Dumbser M, Käser M, Titarev VA Toro EF. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 2007;226:204–43] to ensure monotonicity at discontinuities, and second, a local space–time Galerkin scheme to predict the evolution of the reconstructed polynomial data inside each element during one time step to obtain a high order accurate one-step time discretization. This approach is also able to deal with stiff source terms as shown in [Dumbser M, Enaux C, Toro EF. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J Comput Phys 2008;227:3971–4001]. These two key ingredients are combined with the recently developed path-conservative methods of Parés [Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21] and Castro et al. [Castro MJ, Gallardo JM, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math Comput 2006;75:1103–34] to treat the non-conservative products properly. We show applications of our method to the two-layer shallow water equations as well as applications to the recently published depth-averaged two-fluid flow model of Pitman and Le [Pitman EB, Le L. A two-fluid model for avalanche and debris flows. Philos Trans Roy Soc A 2005;363:1573–601].     

Well-balanced RKDG2 solutions to the shallow water equations over irregular domains with wetting and drying

This paper presents a new one-dimensional (1D) second-order Runge–Kutta discontinuous Galerkin (RKDG2) scheme for shallow flow simulations involving wetting and drying over complex domain topography. The shallow water equations that adopt water level (instead of water depth) as a flow variable are solved by an RKDG2 scheme to give piecewise linear approximate solutions, which are locally defined by an average coefficient and a slope coefficient. A wetting and drying technique proposed originally for a finite volume MUSCL scheme is revised and implemented in the RKDG2 solver. Extra numerical enhancements are proposed to amend the local coefficients associated with water level and bed elevation in order to maintain the well-balanced property of the RKDG2 scheme for applications with wetting and drying. Friction source terms are included and evaluated using splitting implicit discretization, implemented with a physical stopping condition to ensure stability. Several steady and unsteady benchmark tests with/without friction effects are considered to demonstrate the performance of the present model.     

A Robust Reconstruction for Unstructured WENO Schemes

The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order numerical methods for hyperbolic partial differential equations (PDEs). While WENO schemes on structured meshes are quite mature, the development of finite volume WENO schemes on unstructured meshes is more difficult. A major difficulty is how to design a robust WENO reconstruction procedure to deal with distorted local mesh geometries or degenerate cases when the mesh quality varies for complex domain geometry. In this paper, we combine two different WENO reconstruction approaches to achieve a robust unstructured finite volume WENO reconstruction on complex mesh geometries. Numerical examples including both scalar and system cases are given to demonstrate stability and accuracy of the scheme.     

Robust Finite Volume Schemes for Two-Fluid Plasma Equations

Two-fluid plasma equations are derived by taking moments of Boltzmann equations. Ignoring collisions and viscous terms and assuming local thermodynamic equilibrium we get five moment equations for each species (electrons and ions), known as two-fluid plasma equations. These equations allow different temperatures and velocities for electrons and ions, unlike ideal magnetohydrodynamics equations. In this article, we present robust second order MUSCL schemes for two-fluid plasma equations based on Strang splitting of the flux and source terms. The source is treated both explicitly and implicitly. These schemes are shown to preserve positivity of the pressure and density. In the case of explicit treatment of source term, we derive explicit condition on the time step for it to be positivity preserving. The implicit treatment of the source term is shown to preserve positivity, unconditionally. Numerical experiments are presented to demonstrate the robustness and efficiency of these schemes.     

Discontinuous Galerkin Spectral/hp Element Modelling of Dispersive Shallow Water Systems

Two-dimensional shallow water systems are frequently used in engineering practice to model environmental flows. The benefit of these systems are that, by integration over the water depth, a two-dimensional system is obtained which approximates the full three-dimensional problem. Nevertheless, for most applications the need to propagate waves over many wavelengths means that the numerical solution of these equations remains particularly challenging. The requirement for an accurate discretization in geometrically complex domains makes the use of spectral/hp elements attractive. However, to allow for the possibility of discontinuous solutions the most natural formulation of the system is within a discontinuous Galerkin (DG) framework. In this paper we consider the unstructured spectral/hp DG formulation of (i) weakly nonlinear dispersive Boussinesq equations and (ii) nonlinear shallow water equations (a subset of the Boussinesq equations). Discretization of the Boussinesq equations involves resolving third order mixed derivatives. To efficiently handle these high order terms a new scalar formulation based on the divergence of the momentum equations is presented. Numerical computations illustrate the exponential convergence with regard to expansion order and finally, we simulate solitary wave solutions.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case

A class of fifth-order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving one dimensional non-linear hyperbolic conservation law systems, was developed and applied as limiters for the Runge-Kutta discontinuous Galerkin (RKDG) methods in [J. Comput. Phys. 193 (2003) 115]. In this paper, we extend the method to solve two dimensional non-linear hyperbolic conservation law systems. The emphasis is again on the application of such HWENO finite volume methodology as limiters for RKDG methods to maintain compactness of RKDG methods. Numerical experiments for two dimensional Burgers’ equation and Euler equations of compressible gas dynamics are presented to show the effectiveness of these methods.     

Energy Stability Analysis of Some Fully Discrete Numerical Schemes for Incompressible Navier–Stokes Equations on Staggered Grids

In this paper we consider the energy stability estimates for some fully discrete schemes which both consider time and spatial discretizations for the incompressible Navier–Stokes equations. We focus on three kinds of fully discrete schemes, i.e., the linear implicit scheme for time discretization with the finite difference method (FDM) on staggered grids for spatial discretization, pressure-correction schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations, and pressure-stabilization schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations. The energy stability estimates are obtained for the above each fully discrete scheme. The upwind scheme is used in the discretization of the convection term which plays an important role in the design of unconditionally stable discrete schemes. Numerical results are given to verify the theoretical analysis.