A Survey of High Order Schemes for the Shallow Water Equations

In this paper, we survey our recent work on designing high order positivity-preserving well-balanced finite difference and finite volume WENO (weighted essentially non-oscillatory) schemes, and discontinuous Galerkin finite element schemes for solving the shallow water equations with a non-flat bottom topography. These schemes are genuinely high order accurate in smooth regions for general solutions, are essentially non-oscillatory for general solutions with discontinuities, and at the same time they preserve exactly the water at rest or the more general moving water steady state solutions. A simple positivity-preserving limiter, valid under suitable CFL condition, has been introduced in one dimension and reformulated to two dimensions with triangular meshes, and we prove that the resulting schemes guarantee the positivity of the water depth.     

High order finite difference methods on non-uniform meshes for space fractional operators

In the past decades, the finite difference methods for space fractional operators develop rapidly; to the best of our knowledge, all the existing finite difference schemes, including the first and high order ones, just work on uniform meshes. The nonlocal property of space fractional operator makes it difficult to design the finite difference scheme on non-uniform meshes. This paper provides a basic strategy to derive the first and high order discretization schemes on non-uniform meshes for fractional operators. And the obtained first and second schemes on non-uniform meshes are used to solve space fractional diffusion equations. The error estimates and stability analysis are detailedly performed; and extensive numerical experiments confirm the theoretical analysis or verify the convergence orders.     

Two-level finite difference scheme of improved accuracy order for time-dependent problems of mathematical physics

In the theory of finite difference schemes, the most complete results concerning the accuracy of approximate solutions are obtained for two- and three-level finite difference schemes that converge with the first and second order with respect to time. When the Cauchy problem is numerically solved for a system of ordinary differential equations, higher order methods are often used. Using a model problem for a parabolic equation as an example, general requirements for the selection of the finite difference approximation with respect to time are discussed. In addition to the unconditional stability requirements, extra performance criteria for finite difference schemes are presented and the concept of SM stability is introduced. Issues concerning the computational implementation of schemes having higher approximation orders are discussed. From the general point of view, various classes of finite difference schemes for time-dependent problems of mathematical physics are analyzed.     

Development of a high order discretization scheme for solving fully nonlinear magnetohydrodynamic equations

We have developed fully fourth order accurate compact finite difference discretization scheme for the Navier-Stokes equations coupled with Maxwell''s equations. The implementation is done in cylindrical polar geometry. Due to the full-MHD modeling of physical flow, the modeled equations are fully nonlinear coupled hydrodynamic equations which are again coupled with Maxwells equations. In our computations, we have accounted for the induced magnetic field in the flow of an electrically conducting fluid in an external magnetic field. The code is tested against available experimental and theoretical data where applicable. It is observed that a smaller grid of $64 \times 64$ is sufficient for weakly nonlinear problems and higher grids up to $512 \times 512$ are needed as the degree of nonlinearities grow in the modeled equation. In the absence of magnetic field, a discontinuity of total drag coefficient and separation length is noted for $Re=73$ which is in agreement with literature. When the magnetic Reynolds number $Rm<1$ separation length decreases linearly with strength of magnetic field on a log-log scale whereas if $Rm>1$, it decreases nonlinearly, at a much faster rate. Thermal boundary layer thickness decreases as the strength of magnetic field increases and it forces the thermal convection to take place in a laminar structure as observed from thermal contour lines. Finally, using divided differences, we establish that the accuracy of the proposed numerical scheme is in fact fourth order.     

Efficient splitting schemes for magneto-hydrodynamic equations

We present in this paper several efficient numerical schemes for the magneto-hydrodynamic (MHD) equations. These semi-discretized (in time) schemes are based on the standard and rotational pressure-correction schemes for the Navier-Stokes equations and do not involve a projection step for the magnetic field. We show that these schemes are unconditionally energy stable, present an effective algorithm for their fully discrete versions and carry out demonstrative numerical experiments.     

A finite-element coarse-grid projection method for incompressible flow simulations

Coarse grid projection (CGP) methodology is a novel multigrid method for systems involving decoupled nonlinear evolution equations and linear elliptic Poisson equations. The nonlinear equations are solved on a fine grid and the linear equations are solved on a corresponding coarsened grid. Mapping operators execute data transfer between the grids. The CGP framework is constructed upon spatial and temporal discretization schemes. This framework has been established for finite volume/difference discretizations as well as explicit time integration methods. In this article we present for the first time a version of CGP for finite element discretizations, which uses a semi-implicit time integration scheme. The mapping functions correspond to the finite-element shape functions. With the novel data structure introduced, the mapping computational cost becomes insignificant. We apply CGP to pressure-correction schemes used for the incompressible Navier-Stokes flow computations. This version is validated on standard test cases with realistic boundary conditions using unstructured triangular meshes. We also pioneer investigations of the effects of CGP on the accuracy of the pressure field. It is found that although CGP reduces the pressure field accuracy, it preserves the accuracy of the pressure gradient and thus the velocity field, while achieving speedup factors ranging from approximately 2 to 30. The minimum speedup occurs for velocity Dirichlet boundary conditions, while the maximum speedup occurs for open boundary conditions.     

Boosting the accuracy of finite difference schemes via optimal time step selection and non-iterative defect correction

In this article, we present a unified analysis of the simple technique for boosting the order of accuracy of finite difference schemes for time dependent partial differential equations (PDEs) by optimally selecting the time step used to advance the numerical solution and adding defect correction terms in a non-iterative manner. The power of the technique, which is applicable to time dependent, semilinear, scalar PDEs where the leading-order spatial derivative has a constant coefficient, is its ability to increase the accuracy of formally low-order finite difference schemes without major modification to the basic numerical algorithm. Through straightforward numerical analysis arguments, we explain the origin of the boost in accuracy and estimate the computational cost of the resulting numerical method. We demonstrate the utility of optimal time step (OTS) selection combined with non-iterative defect correction (NIDC) on several different types of finite difference schemes for a wide array of classical linear and semilinear PDEs in one and more space dimensions on both regular and irregular domains.     

Finite differences and collocation methods for the solution of the two‐dimensional heat equation

In this article, we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two‐dimensional heat equation. We employ, respectively, second‐order and fourth‐order schemes for the spatial derivatives, and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is nonsingular. Numerical experiments carried out on serial computers show the unconditional stability of the proposed method and the high accuracy achieved by the fourth‐order scheme. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 54–63, 2001     

Second order unconditionally convergent and energy stable linearized scheme for MHD equations

In this paper, we propose an efficient numerical scheme for magnetohydrodynamics (MHD) equations. This scheme is based on a second order backward difference formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms. Meanwhile, the mixed finite element method is used for spatial discretization. We present that the scheme is unconditionally convergent and energy stable with second order accuracy with respect to time step. The optimal L ² and H ¹ fully discrete error estimates for velocity, magnetic variable and pressure are also demonstrated. A series of numerical tests are carried out to confirm our theoretical results. In addition, the numerical experiments also show the proposed scheme outperforms the other classic second order schemes, such as Crank-Nicolson/Adams-Bashforth scheme, linearized Crank-Nicolson’s scheme and extrapolated Gear’s scheme, in solving high physical parameters MHD problems.     

THE FINITE DIFFERENCE STREAMLINE DIFFUSION METHODS FOR TIME-DEPENDENT CONVECTION-DIFFUSION EQUATIONS

In this paper, two finite difference streamline diffusion (FDSD) schemes for solving two-dimensional time-dependent convection-diffusion equations are constructed. Stability and optimal order error estimati-ions for considered schemes are derived in the norm stronger than L~2-norm.     

High order Hybrid central—WENO finite difference scheme for conservation laws

In this article we present a high resolution hybrid central finite difference—WENO scheme for the solution of conservation laws, in particular, those related to shock–turbulence interaction problems. A sixth order central finite difference scheme is conjugated with a fifth order weighted essentially non-oscillatory WENO scheme in a grid-based adaptive way. High order multi-resolution analysis is used to detect the high gradients regions of the numerical solution in order to capture the shocks with the WENO scheme while the smooth regions are computed with the more efficient and accurate central finite difference scheme. The application of high order filtering to mitigate the dispersion error of central finite difference schemes is also discussed. Numerical experiments with the 1D compressible Euler equations are shown.     

A study on a second order finite difference scheme for fractional advection–diffusion equations

Second order finite difference schemes for fractional advection–diffusion equations are considered in this paper. We note that, when studying these schemes, advection terms with coefficients having the same sign as those of diffusion terms need additional estimates. In this paper, by comparing generating functions of the corresponding discretization matrices, we find that sufficiently strong diffusion can dominate the effects of advection. As a result, convergence and stability of schemes are obtained in this situation.     

Analysis on two approaches for high order accuracy finite difference computation

We analyze two approaches for enhancing the accuracy of the standard second order finite difference schemes in solving one dimensional elliptic partial differential equations. These are the fourth order compact difference scheme and the fourth order scheme based on the Richardson extrapolation techniques. We study the truncation errors of these approaches and comment on their regularity requirements and computational costs. We present numerical experiments to demonstrate the validity of our analysis.     

ANTI-DIFFUSIVE FINITE DIFFERENCE WENO METHODS FOR SHALLOW WATER WITH TRANSPORT OF POLLUTANT

In this paper we further explore and apply our recent anti-diffusive flux corrected highorder finite difference WENO schemes for conservation laws [18] to compute the Saint-Venant system of shallow water equations with pollutant propagation, which is describedby a transport equation. The motivation is that the high order anti-diffusive WENOscheme for conservation laws produces sharp resolution of contact discontinuities whilekeeping high order accuracy for the approximation in the smooth region of the solution.The application of the anti-diffusive high order WENO scheme to the Saint-Venant systemof shallow water equations with transport of pollutant achieves high resolution     

A minimum entropy principle of high order schemes for gas dynamics equations

The entropy solutions of the compressible Euler equations satisfy a minimum principle for the specific entropy (Tadmor in Appl Numer Math 2:211–219, 1986). First order schemes such as Godunov-type and Lax-Friedrichs schemes and the second order kinetic schemes (Khobalatte and Perthame in Math Comput 62:119–131, 1994) also satisfy a discrete minimum entropy principle. In this paper, we show an extension of the positivity-preserving high order schemes for the compressible Euler equations in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and Zhang et?al. (J Scientific Comput, in press), to enforce the minimum entropy principle for high order finite volume and discontinuous Galerkin (DG) schemes.     

Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

The main objective of this paper is optimization of second‐order finite difference schemes for elliptic equations, in particular, for equations with singular solutions and exterior problems. A model problem corresponding to the Laplace equation on a semi‐infinite strip is considered. The boundary impedance (Neumann‐to‐Dirichlet map) is computed as the square root of an operator using the standard three‐point finite difference scheme with optimally chosen variable steps. The finite difference approximation of the boundary impedance for data of given smoothness is the problem of rational approximation of the square root on the operator's spectrum. We have implemented Zolotarev's optimal rational approx‐imant obtained in terms of elliptic functions. We have also found that a geometrical progression of the grid steps with optimally chosen parameters is almost as good as the optimal approximant. For bounded operators it increases from second to exponential the convergence order of the finite difference impedance with the convergence rate proportional to the inverse of the logarithm of the condition number. For the case of unbounded operators in Sobolev spaces associated with elliptic equations, the error decays as the exponential of the square root of the mesh dimension. As an example, we numerically compute the Green function on the boundary for the Laplace equation. Some features of the optimal grid obtained for the Laplace equation remain valid for more general elliptic problems with variable coefficients. © 2000 John Wiley & Sons, Inc.     

Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients

We present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients. Based on the Maple software package, we approximate the leading terms in the truncation error of the Taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation. A test problem is solved numerically to validate the derived fourth order compact difference scheme. This symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations.     

On stability of numerical schemes via frozen coefficients and the magnetic induction equations

We study finite difference discretizations of initial boundary value problems for linear symmetric hyperbolic systems of equations in multiple space dimensions. The goal is to prove stability for SBP-SAT (Summation by Parts—Simultaneous Approximation Term) finite difference schemes for equations with variable coefficients. We show stability by providing a proof for the principle of frozen coefficients, i.e., showing that variable coefficient discretization is stable provided that all corresponding constant coefficient discretizations are stable.     

Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations

In this paper, some reduced finite difference schemes based on a proper orthogonal decomposition (POD) technique for parabolic equations are derived. Also the error estimates between the POD approximate solutions of the reduced finite difference schemes and the exact solutions for parabolic equations are established. It is shown by considering the results of two numerical examples that the numerical results are consistent with theoretical conclusions. Moreover, it is also shown that the POD reduced finite difference schemes are feasible and efficient.