High-resolution central difference scheme for the shallow water equations

A two-dimensional nonoscillatory central difference scheme was extended to the shallow water equations. A high-resolution numerical method for solving the shallow water equations was presented. In order to prevent oscillation, the nonlinear limiter is employed to approximate the discrete slopes. The main advantage of the presented method is simplicity comparable with the upwind schemes. This method does not require Riemann solvers or some form of flux difference splitting methods. Furthermore, the discrete derivatives of flux can be approximated by the component-wise approach and thus the computation of Jacobian can be avoided. The method retains high resolution and high accuracy similar to the upwind results. It is applied to simulating several tests, including circular dam-break problem, shock focusing problem and partial dam-break problem. The results are in good agreement with the numerical results obtained by other methods. The simulated results also demonstrate that the presented method is stable and efficient.

Comments on Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

The explicit compact difference scheme, proposed in Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD by Lin et al., published in Applied Mathematics and Mechanics （English Edition）, 2007, 28（7）, 943-953, has the same performance as the conventional finite difference schemes. It is just another expression of the conventional finite difference schemes. The proposed expression does not have the advantages of a compact difference scheme. Nonetheless, we can more easily obtain and implement compared with the conventional expression in which the coefficients can only be obtained by solving equations, especially for higher accurate schemes.

Revisting High-Resolution Schemes with van Albada Slope Limiter

Slope limiters play an essential role in maintaining the non-oscillatory behavior of high-resolution methods for nonlinear conservation laws.The family of minmod limiters serves as the prototype example.Here,we revisit the question of non-oscillatory behavior of high-resolution central schemes in terms of the slope limiter proposed by van Albada et al.(Astron Astrophys 108:76–84,1982).The van Albada(vA)limiter is smoother near extrema,and consequently,in many cases,it outperforms the results obtained using the standard minmod limiter.In particular,we prove that the vA limiter ensures the one-dimensional Total-Variation Diminishing(TVD)stability and demonstrate that it yields noticeable improvement in computation of one-and two-dimensional systems.

UNSTEADY/STEADY NUMERICAL SIMULATION OF THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON ARTIFICIAL COMPRESSIBILITY

A mixed algorithm of central and upwind difference scheme for the solution of steady/unsteady incompressible Navier-Stokes equations is presented. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The numerical flux of semi-discrete equations is computed by using the Roe approximation. Time accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The algebraic turbulence model of Baldwin-Lomax is ulsed in this work. As examples, the solutions of flow through two dimensional flat, airfoil, prolate spheroid and cerebral aneurysm are computed and the results are compared with experimental data. The results show that the coefficient of pressure and skin friction are agreement with experimental data, the largest discrepancy occur in the separation region where the lagebraic turbulence model of Baldwin-Lomax could not exactly predict the flow.

A numerical study of 1-D nonlinear P-wave propagation in solid

Two central schemes of finite difference (FD) up to different accuracy orders of space sampling step Dx (Fourth order and Sixth order respectively) were used to study the 1-D nonlinear P-wave propagation in the nonlinear solid media by the numerical method. Distinctly different from the case of numerical modeling of linear elastic wave, there may be several difficulties in the numerical treatment to the nonlinear partial differential equation, such as the steep gradients, shocks and unphysical oscillations. All of them are the great obstacles to the stability and conver-gence of numerical calculation. Fortunately, the comparative study on the modeling of nonlinear wave by the two FD schemes presented in the paper can provide us with an easy method to keep the stability and convergence in the calculation field when the product of the absolute value of nonlinear coefficient and the value of u/x are small enough, namely, the value of bu/x is much smaller than 1. Several results are founded in the numerical study of nonlinear P-wave propagation, such as the waveform aberration, the generation and growth of harmonic wave and the energy redistribution among different frequency components. All of them will be more violent when the initial amplitude A0 is larger or the nonlinearity of medium is stronger. Correspondingly, we have found that the nonlinear P-wave propagation velocity will change with different initial frequency f of source wave or the wave velocity c (equal to the P-wave velocity in the same medium without considering nonlinearity).

Multidomain Hybrid Direct DG and Central Difference Methods for Viscous Terms in Hyperbolic-Parabolic Equations

A class of multidomain hybrid methods of direct discontinuous Galerkin(DDG)methods and central difference(CD)schemes for the viscous terms is pro-posed in this paper.Both conservative and nonconservative coupling modes are dis-cussed.To treat the shock wave,the nonconservative coupling mode automatically switch to conservative coupling mode to preserve the conservative property when discontinuities pass through the artificial interface.To maintain the accuracy of the hybrid methods,the Lagrange interpolation polynomials and their derivatives are reconstructed to handle the coupling cells in the DDG subdomain,while the values of ghost points for the CD subdomain are calculated by the approximate polynomials from the DDG methods.The linear stabilities of these methods are demonstrated in detail through von-Neumann analysis.The multidomain hybrid DDG and CD meth-ods are then extended to one-and two-dimensional hyperbolic-parabolic equations.Numerical results validate that the multidomain hybrid methods are high-order ac-curate in the smooth regions,robust for viscous shock simulations and capable to save computational cost.

SOLUTION OF 2D SHALLOW WATER EQUATIONS BY GENUINELY MULTIDIMENSIONAL SEMI-DISCRETE CENTRAL SCHEME

A numerical two-dimensional shallow water method was based on method for solving the equations was presented. This the third-order genuinely multidimensional semi-discrete central scheme for spatial discretization and the optimal third-order Strong Stability Preserving （SSP） Runge-Kutta method for time integration. The third-order compact Central Weighted Essentially Non-Oscillatory （CWENO） reconstruction was adopted to guarantee the non-oscillatory behavior of the presented scheme and improve the resolution. Two kinds of source terms were considered in this work. They were evaluated using different approaches. The resulting scheme does not require Riemann solvers or characteristic decomposition, hence it retains all the attractive features of central schemes such as simplicity and high resolution. To evaluate the performance of the presented scheme, several numerical examples were tested. The results demonstrate that our method is efficient, stable and robust.

Non-Oscillatory Hierarchical Reconstruction for Central and Finite Volume Schemes

This is the continuation of the paper”Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction”by the same authors.The hierarchical reconstruction introduced therein is applied to central schemes on overlapping cells and to finite volume schemes on non-staggered grids.This takes a new finite volume approach for approximating non-smooth solutions.A critical step for high-order finite volume schemes is to reconstruct a non-oscillatory high degree polynomial approximation in each cell out of nearby cell averages.In the paper this procedure is accomplished in two steps:first to reconstruct a high degree polynomial in each cell by using e.g.,a central reconstruction,which is easy to do despite the fact that the reconstructed polynomial could be oscillatory;then to apply the hierarchical reconstruction to remove the spurious oscillations while maintaining the high resolution.All numerical computations for systems of conservation laws are performed without characteristic decomposition.In particular,we demonstrate that this new approach can generate essentially non-oscillatory solutions even for 5th-order schemes without characteristic decomposition.

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.

A CHARACTERISTIC-BASED FINITE VOLUME SCHEME FOR SHALLOW WATER EQUATIONS

We propose a new characteristic-based finite volume scheme combined with the method of Central Weighted Essentially Non-Oscillatory （CWENO） reconstruction and characteristics, to solve shallow water equations. We apply the scheme to simulate dam-break problems. A number of challenging test cases are considered, such as large depth differences even wet/dry bed. The numerical solutions well agree with the analytical solutions. The results demonstrate the desired accuracy, high-resolution and robustness of the presented scheme.

Well-Balanced Central Scheme for the System of MHD Equations with Gravitational Source Term

A well-balanced second order finite volume central scheme for the magnetohydrodynamic(MHD)equations with gravitational source term is developed in this paper.The scheme is an unstaggered central scheme that evolves the numerical solution on a single grid and avoids solving Riemann problems at the cell interfaces using ghost staggered cells.A subtraction technique is used on the conservative variables with the support of a known steady state in order to manifest the well-balanced property of the scheme.The divergence-free constraint of themagnetic field is satisfied after applying the constrained transport method(CTM)for unstaggered central schemes at the end of each time-step by correcting the components of the magnetic field.The robustness of the proposed scheme is verified on a list of numerical test cases from the literature.

A New Fifth-Order Finite Volume Central WENO Scheme for Hyperbolic Conservation Laws on Staggered Meshes

In this paper,a new fifth-order finite volume central weighted essentially non-oscillatory(CWENO)scheme is proposed for solving hyperbolic conservation laws on staggered meshes.The high-order spatial reconstruction procedure using a convex combination of a fourth degree polynomial with two linear polynomials(in one dimension)or four linear polynomials(in two dimensions)in a traditional WENO fashion and a time discretization method using the natural continuous extension(NCE)of the Runge-Kutta method are applied to design this new fifth-order CWENO scheme.This new finite volume CWENO scheme uses the information defined on the same largest spatial stencil as that of the same order classical CWENO schemes[37,46]with the application of smaller number of unequal-sized spatial stencils.Since the new nonlinear weights are adopted,the new finite volume CWENO scheme could obtain the same order of accuracy and get smaller truncation errors in L1 and L¥norms in smooth regions,and control the spurious oscillations near strong shocks or contact discontinuities.The new CWENO scheme has advantages over the classical CWENO schemes[37,46]on staggered meshes in its simplicity and easy extension to multi-dimensions.Some one-dimensional and two-dimensional benchmark numerical examples are provided to illustrate the good performance of this new fifthorder finite volume CWENO scheme.

Central Schemes and Second Order Boundary Conditions for 1D Interface and Piston Problems in Lagrangian Coordinates

We study high-resolution central schemes in Lagrangian coordinates for the one-dimensional system of conservation laws describing the evolution of two gases in slab geometry separated by an interface.By using Lagrangian coordinates,the interface is transformed to a fixed coordinate in the computational domain and,as a consequence,the movement of the interface is obtained as a byproduct of the numerical solution.The main contribution is the derivation of a special equation of state to be imposed at the interface in order to avoid non-physical oscillations.Suitable boundary conditions at the piston that guarantee second order convergence are described.We compare the solution of the piston problem to other results available in the literature and to a reference solution obtained within the adiabatic approximation.A shock-interface interaction problem is also treated.The results on these tests are in good agreement with those obtained by other methods.

Response of a spherical cavity in a fully-coupled thermo-poro-elastodynamic medium by cell-adaptive second-order central high resolution schemes

In this study,fully coupled thermo-poroelastic saturated media are simulated by a grid/cell adaptive central high resolution scheme.The central method corresponds to the second order Kurganov-Tadmor(KT)scheme working on adapted cells with the total variation diminishing(TVD)stability condition.The coupled equations include motion,fluid flow,heat flow,continuity condition,and a constitutive equation.The grid/cell adaptation is performed by the interpolating wavelet transform in the multiresolution framework to capture fine scale responses and to obtain a computationally effective solver.With respect to the use of central schemes,the coupled equations should be re-expressed as a system of coupled first-order hyperbolic-parabolic partial differential equations(PDEs)with possible source(load)terms.The system is initially derived in the Cartesian coordinate system,and it is subsequently modified to consider a spherical cavity in isotropic,symmetric,and saturated media in the spherical coordinate system.It is assumed that the cavity boundary is subjected to sudden time-dependent thermal/mechanical sources.Discontinuous propagating fronts develop in the media due to the aforementioned loading.It is challenging to handle these solutions with numerical methods,and special attention is required to prevent/control numerical dispersion and dissipation.Hence,as previously mentioned,adaptive central high resolution schemes are employed in the present study.

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.

A well-balanced positivity preserving two-dimensional shallow flow model with wetting and drying fronts over irregular topography

This paper presents an improved well-balanced Godunov-type 2-D finite volume model with structured grids to simulate shallow flows with wetting and drying fronts over an irregular topography. The intercell flux is computed using a central upwind scheme, which is a Riemann-problem-solver-free method for hyperbolic conservation laws. The nonnegative reconstruction method for the water depth is implemented to resolve the stationary or wet/dry fronts. The bed slope source term is discretized using a central difference method to capture the static flow state over the irregular topography. Second-order accuracy in space is achieved by using the slope limited linear reconstruction method. With the proposed method, the model can avoid the partially wetting/drying cell problem and maintain the mass conservation. The proposed model is tested and verified against three theoretical benchmark tests and two experimental dam break flows. Further, the model is applied to predict the maximum water level and the flood arrival time at different gauge points for the Malpasset dam break event. The predictions agree well with the numerical results and the measurement data published in literature, which demonstrates that with the present model, a well-balanced state can be achieved and the water depth can be nonnegative when the Courant number is kept less than 0.25.

Two-Way Interactions Between Equatorially-Trapped Waves and the Barotropic Flow

Lateral energy exchange between the tropics and the midlatitudes is a topic of great importance for understanding Earth＇s climate system. In this paper, the authors address this issue in an idealized set up through simple shallow water models for the interactions between equatorially trapped waves and the barotropic mode, which supports Rossby waves that propagate poleward and can excite midlatitude teleconnection patterns. It is found here that the interactions between a Kelvin wave and a fixed meridionai shear （mimicking the jet stream） generates a non-trivial meridional velocity and meridional convergence in phase with the upward motion that can attain a maximum of about 50%, which oscillates on frequencies ranging from one day to 10 days. When, on the other hand, the barotropic flow is forced by slowly propagating Kelvin waves a complex flow pattern emerges; it consists of a phase-locked barotropic response that is equatoriaily trapped and that propagates eastward with the forcing Kelvin wave and a certain number of planetary Rossby waves that propagate westward and toward the poles as seen in nature. It is suggested here that the poleward propagating waves are to some sort of multi-way resonant interaction with the phase locked response. Moreover, it is shown here that a numerical scheme with dispersion properties that depend on the direction perpendicular to the direction of propagation, namely the 2D central scheme of Nessyahu and Tadmor, can artificially alter significantly the topology of the wave fields and thus should be avoided in climate models.

The Space-Time CE/SE Method for Solving Reduced Two-Fluid Flow Model

The space-time conservation element and solution element(CE/SE)method is proposed for solving a conservative interface-capturing reducedmodel of compressible two-fluid flows.The flow equations are the bulk equations,combined with mass and energy equations for one of the two fluids.The latter equation contains a source term for accounting the energy exchange.The one and two-dimensional flow models are numerically investigated in this manuscript.The CE/SE method is capable to accurately capture the sharp propagating wavefronts of the fluids without excessive numerical diffusion or spurious oscillations.In contrast to the existing upwind finite volume schemes,the Riemann solver and reconstruction procedure are not the building block of the suggested method.The method differs from the previous techniques because of global and local flux conservation in a space-time domain without resorting to interpolation or extrapolation.In order to reveal the efficiency and performance of the approach,several numerical test cases are presented.For validation,the results of the current method are compared with other finite volume schemes.

Operator Splitting for Three-Phase Flow in Heterogeneous Porous Media

We describe an operator splitting technique based on physics rather than on dimension for the numerical solution of a nonlinear system of partial differential equations which models three-phase flow through heterogeneous porous media.The model for three-phase flow considered in this work takes into account capillary forces,general relations for the relative permeability functions and variable porosity and permeability fields.In our numerical procedure a high resolution,nonoscillatory,second order,conservative central difference scheme is used for the approximation of the nonlinear system of hyperbolic conservation laws modeling the convective transport of the fluid phases.This scheme is combined with locally conservative mixed finite elements for the numerical solution of the parabolic and elliptic problems associated with the diffusive transport of fluid phases and the pressure-velocity problem.This numerical procedure has been used to investigate the existence and stability of nonclassical shock waves(called transitional or undercompressive shock waves)in two-dimensional heterogeneous flows,thereby extending previous results for one-dimensional flow problems.Numerical experiments indicate that the operator splitting technique discussed here leads to computational efficiency and accurate numerical results.