Semi-discrete Central Scheme Research Articles

A central conservative scheme for general rectangular grids

We present an extension of the genuinely multi-dimensional semi-discrete central scheme developed in [A. Kurganov, S. Noelle, G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, SIAM J. Sci. Comput. 23 (3) (2001) 707–740.] to arbitrary orthogonal grids. The presented algorithm is constructed to yield the geometric scaling factors in a self-consistent way. Additionally, the order of the scheme is not fixed during the derivation of the basic algorithm. Based on the resulting general scheme it is possible to construct methods of any desired order, just by considering the corresponding reconstruction polynomial. We demonstrate how a second order scheme in plane polar coordinates and cylindrical coordinates can be derived from our general formulation. Finally, we demonstrate the correctness of this second order scheme through application to several numerical experiments.

A multidimensional magnetohydrodynamics code based on semidiscrete central WENO and PPM schemes

We present a multidimensional magnetohydrodynamics code based on semidiscrete central Godunov-type schemes and high order weighted essentially non-oscillatory data reconstruction and incorporate elements of the piecewise parabolic method into the reconstruction schemes. This algorithm avoids spectral decomposition into characteristic waves and is easily implemented. We have implemented this algorithm and carried out one and two dimensional tests, which verifies that this code can achieve the accuracy and efficiency to solve the astrophysical problems.

SEMI-DISCRETE CENTRAL DIFFERENCE METHOD FOR DETERMINING SURFACE HEAT FLUX OF IHCP

We consider an inverse heat conduction problem(IHCP) in a quarter plane which appears in some applied subjects. We want to determine the heat flux on the surface of a body from a measured temperature history at a fixed location inside the body. This is a severely ill-posed problem in the sense that arbitrarily "small" differences in the input temperature data may lead to arbitrarily "large" differences in the surface flux. A semi-discrete central difference scheme in time is employed to deal with the ill posed problem. We obtain some error estimates which also give the information about how to choose the step length in time. Some numerical examples illustrate the effects of the proposed method.

A low dissipation essentially non-oscillatory central scheme

Here we present a new, semidiscrete, central scheme for the numerical solution of one-dimensional systems of hyperbolic conservation laws. The method presented in this paper is an extension of the centrally weighted non-oscillatory schemes ( Cweno) presented in [A. Kurganov, E. Tadmor, New high resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comp. Phys. 160 (2000) 241–282; A. Kurganov, D. Levy, A third-order semidiscrete central scheme for conservation laws and convection–diffusion equations, SIAM J. Sci. Comput. 22 (2000) 1461–1488; A. Kurganov, S. Noelle, G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, SIAM J. Sci. Comput. 23 (3) (2001) 707–740]. The method suggested in this manuscript is derived independently of the order of the scheme. The gain in this new method is a decreased dissipation especially for high Mach-number flows, which are frequently encountered, e.g., in astrophysical contexts or turbulent systems.

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

A numerical method for solving the two-dimensional shallow water equations was presented. This method was based on 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.

Mapped WENO and weighted power ENO reconstructions in semi-discrete central schemes for Hamilton–Jacobi equations

We incorporate new high-order WENO-type reconstructions into Godunov-type central schemes for Hamilton–Jacobi equations. We study schemes that are obtained by combining the Kurganov–Noelle–Petrova flux with the weighted power ENO and the mapped WENO reconstructions. We also derive new variants of these reconstructions by composing the weighted power ENO and the mapped WENO reconstructions with each other. While all schemes are, formally, fifth-order accurate, we show that the quality of the approximation does depend on the particular reconstruction that is being used. In certain cases, it is shown that the approximate solution may not converge to the viscosity solution at all.

On the Total Variation of High-Order Semi-Discrete Central Schemes for Conservation Laws

We discuss a new fifth-order, semi-discrete, central-upwind scheme for solving one-dimensional systems of conservation laws. This scheme combines a fifth-order WENO reconstruction, a semi-discrete central-upwind numerical flux, and a strong stability preserving Runge---Kutta method. We test our method with various examples, and give particular attention to the evolution of the total variation of the approximations.

Nonoscillatory Central Schemes for One- and Two-Dimensional Magnetohydrodynamics Equations. II: High-Order SemiDiscrete Schemes

We present a new family of high-resolution, nonoscillatory semidiscrete central schemes for the approximate solution of the ideal magnetohydrodynamics (MHD) equations. This is the second part of our work, where we are passing from the fully discrete staggered schemes in [J. Balbás, E. Tadmor, and C.-C. Wu, J. Comput. Phys., 201 (2004), pp. 261-285] to the semidiscrete formulation advocated in [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241-282]. This semidiscrete formulation retains the simplicity of fully discrete central schemes while enhancing efficiency and adding versatility. The semidiscrete algorithm offers a wider range of options to implement its two key steps: nonoscillatory reconstruction of point values followed by the evolution of thecorresponding point valued fluxes. We present the solution of several prototype MHD problems. Solutions of one-dimensional Brio--Wu shock-tube problems and the two-dimensional Kelvin--Helmholtz instability, Orszag--Tang vortex system, and the disruption of a high density cloud by a strong shock are carried out using third- and fourth-order central schemes based on the central WENO reconstructions. These results complement those presented in our earlier work and confirm the remarkable versatility and simplicity of central schemes as black-box, Jacobian-free MHD solvers. Furthermore, our numerical experiments demonstrate that this family of semidiscrete central schemes preserves the $\nabla \cdot {\bf B} = 0$-constraint within machine round-off error; happily, no constrained-transport enforcement is needed.

Central WENO Schemes for Hamilton–Jacobi Equations on Triangular Meshes

We derive Godunov‐type semidiscrete central schemes for Hamilton–Jacobi equations on triangular meshes. High‐order schemes are then obtained by combining our new numerical fluxes with high‐order WENO reconstructions on triangular meshes. The numerical fluxes are shown to be monotone in certain cases. The accuracy and high‐resolution properties of our scheme are demonstrated in a variety of numerical examples.

A stable semi-discrete central scheme for the two-dimensional incompressible Euler equations

We derive a second-order, semi-discrete central-upwind scheme for the incompressible 2D Euler equations in the vorticity formulation. The reconstructed velocity field preserves an exact discrete incompressibility relation. We state a local maximum principle for a fully discrete version of the scheme and prove it using a convexity argument. We then show how similar convexity arguments can be used to prove that the scheme maps certain Orlicz spaces into themselves. The consequences of this result on the convergence of the scheme are discussed. Numerical simulations support the expected properties of the scheme.

A central differencing simulation of the Orszag-Tang vortex system

The Orszag-Tang vortex system describes the transition to supersonic turbulence for the equations of magnetohydrodynamics (MHD) in two space dimensions. The complex interaction between various shock waves traveling at different speed regimes that characterizes the solution of this test problem requires the use of numerical schemes capable of detecting and resolving accurately steep gradients while avoiding the onset of spurious oscillations. A simulation of the Orszag-Tang MHD vortex system computed with a third-order semi-discrete central scheme (Kurganov and Tadmor, 2000), (Balbas and Tadmor, submitted to SIAM Journal of Scientific Computing) is presented below. The central differencing approach avoids any detailed knowledge of the characteristic structure of the hyperbolic model, resulting in simple to implement, yet robust, black-box numerical schemes, (Balbas and Tadmor, 2004).

Semi-discrete central-upwind schemes with reduced dissipation for Hamilton-Jacobi equations

We introduce a new family of Godunov-type semi-discrete central schemes for multidimensional Hamilton-Jacobi equations. These schemes are a less dissipative generalization of the central-upwind schemes that have been recently proposed in Kurganov, Noelle and Petrova (2001, SIAM J. Sci. Comput., 23, pp. 707-740). We provide the details of the new family of methods in one, two, and three space dimensions, and then verify their expected low-dissipative property in a variety of examples.

Central-Upwind Schemes for the Saint-Venant System

We present one- and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very dicult to compute. Our approach is based on extending semi-discrete central schemes for systems of hyperbolic conservation laws to balance laws. Special attention is paid to the discretization of the source term such as to preserve stationary steady-state solutions. We also prove that the second-order version of our schemes preserves the nonnegativity of the height of the water. This important feature allows one to compute solutions for problems that include dry areas.

Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton--Jacobi Equations

We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton--Jacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the schemes from [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241--282; A. Kurganov and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461--1488; A. Kurganov and G. Petrova, A third-order semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., to appear] and [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 720--742]. The main advantages of the proposed central schemes are the high resolution, due to the smaller amount of the numerical dissipation, and the simplicity. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide variety of applications. At the same time, the developed schemes have an upwind nature, since they respect the directions of wave propagation by measuring the one-sided local speeds. This is why we call them central-upwind schemes. The constructed schemes are applied to various problems, such as the Euler equations of gas dynamics, the Hamilton--Jacobi equations with convex and nonconvex Hamiltonians, and the incompressible Euler and Navier--Stokes equations. The incompressibility condition in the latter equations allows us to treat them both in their conservative and transport form. We apply to these problems the central-upwind schemes, developed separately for each of them, and compute the corresponding numerical solutions.

New High-Resolution Semi-discrete Central Schemes for Hamilton–Jacobi Equations

We introduce a new high-resolution central scheme for multidimensional Hamilton–Jacobi equations. The scheme retains the simplicity of the non-oscillatory central schemes developed by C.-T. Lin and E. Tadmor (in press, SIAM J. Sci. Comput.), yet it enjoys a smaller amount of numerical viscosity, independent of 1/Δt. By letting Δt↓0 we obtain a new second-order central scheme in the particularly simple semi-discrete form, along the lines of the new semi-discrete central schemes recently introduced by the authors in the context of hyperbolic conservation laws. Fully discrete versions are obtained with appropriate Runge–Kutta solvers. The smaller amount of dissipation enables efficient integration of convection-diffusion equations, where the accumulated error is independent of a small time step dictated by the CFL limitation. The scheme is non-oscillatory thanks to the use of nonlinear limiters. Here we advocate the use of such limiters on second discrete derivatives, which is shown to yield an improved high resolution when compared to the usual limitation of first derivatives. Numerical experiments demonstrate the remarkable resolution obtained by the proposed new central scheme.

New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection–Diffusion Equations

Central schemes may serve as universal finite-difference methods for solving nonlinear convection–diffusion equations in the sense that they are not tied to the specific eigenstructure of the problem, and hence can be implemented in a straightforward manner as black-box solvers for general conservation laws and related equations governing the spontaneous evolution of large gradient phenomena. The first-order Lax–Friedrichs scheme (P. D. Lax, 1954) is the forerunner for such central schemes. The central Nessyahu–Tadmor (NT) scheme (H. Nessyahu and E. Tadmor, 1990) offers higher resolution while retaining the simplicity of the Riemann-solver-free approach. The numerical viscosity present in these central schemes is of order O((Δx)2r/Δt). In the convective regime where Δt∼Δx, the improved resolution of the NT scheme and its generalizations is achieved by lowering the amount of numerical viscosity with increasing r. At the same time, this family of central schemes suffers from excessive numerical viscosity when a sufficiently small time step is enforced, e.g., due to the presence of degenerate diffusion terms.In this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order O(Δx)2r−1)). In particular, our new central schemes maintain their high-resolution independent of O(1/Δt), and letting Δt ↓ 0, they admit a particularly simple semi-discrete formulation. The main idea behind the construction of these central schemes is the use of more precise information of the local propagation speeds. Beyond these CFL related speeds, no characteristic information is required. As a second ingredient in their construction, these central schemes realize the (nonsmooth part of the) approximate solution in terms of its cell averages integrated over the Riemann fans of varying size.The semi-discrete central scheme is then extended to multidimensional problems, with or without degenerate diffusive terms. Fully discrete versions are obtained with Runge–Kutta solvers. We prove that a scalar version of our high-resolution central scheme is nonoscillatory in the sense of satisfying the total-variation diminishing property in the one-dimensional case and the maximum principle in two-space dimensions. We conclude with a series of numerical examples, considering convex and nonconvex problems with and without degenerate diffusion, and scalar and systems of equations in one- and two-space dimensions. Time evolution is carried out by the third- and fourth-order explicit embedded integration Runge–Kutta methods recently proposed by A. Medovikov (1998). These numerical studies demonstrate the remarkable resolution of our new family of central scheme.

A Third-Order Semidiscrete Central Scheme for Conservation Laws and Convection-Diffusion Equations

We present a new third-order, semidiscrete, central method for approximating solutions to multidimensional systems of hyperbolic conservation laws, convection-diffusion equations, and related problems. Our method is a high-order extension of the recently proposed second-order, semidiscrete method in [A. Kurgonov and E. Tadmor, J. Comput Phys., 160 (2000) pp. 241--282]. The method is derived independently of the specific piecewise polynomial reconstruction which is based on the previously computed cell-averages. We demonstrate our results by focusing on the new third-order central weighted essentially nonoscillatory (CWENO) reconstruction presented in [D. Levy, G. Puppo, and G. Russo, SIAM J. Sci. Comput., 21 (1999), pp. 294--322]. The numerical results we present show the desired accuracy, high resolution, and robustness of our method.