High-resolution Central Scheme Research Articles

A Mach-Uniform Strictly Conservative High-Resolution Scheme for Unsteady Compressible Flows of Gases and Liquids

We present a new numerical method to simulate unsteady compressible flows under arbitrary Mach number condition. A hybrid explicit / fractional-step temporal discretization for Euler equation on strictly conservative formulation has been developed. The explicit method is adopted when the flow speed is high so that strong shocks as well as sound waves are accurately captured; while the fractional-step pressure-iteration method is adopted when the flow speed is low so that the CFL condition as a constraint on the time step can be avoided. We selected as the base scheme Jiang et al.'s nonstaggered version of the high-resolution central scheme by Nessyahu and Tadmor for hyperbolic conservation law since the scheme is well-suited to construct the fractional-step algorithm. An additional pressure correction procedure (GCUP) is implemented to deal with general equation of state. Riemann problems in one dimension are solved for a gas and a liquid to show high accuracy and robustness of the present scheme. A flow transition calculation from subsonic to supersonic due to a heat source is also successful.

Magnetohydrodynamics in full general relativity: Formulation and tests

A new implementation for magnetohydrodynamics (MHD) simulations in full general relativity (involving dynamical spacetimes) is presented. In our implementation, Einstein's evolution equations are evolved by a Baumgarte-Shapiro-Shibata-Nakamura formalism, MHD equations by a high-resolution central scheme, and induction equation by a constraint transport method. We perform numerical simulations for standard test problems in relativistic MHD, including special relativistic magnetized shocks, general relativistic magnetized Bondi flow in stationary spacetime, and a long-term evolution for self-gravitating system composed of a neutron star and a magnetized disk in full general relativity. In the final test, we illustrate that our implementation can follow winding-up of the magnetic field lines of magnetized and differentially rotating accretion disks around a compact object until saturation, after which magnetically driven wind and angular momentum transport inside the disk turn on.

Robustness of a high-resolution central scheme for hydrodynamic simulations in full general relativity

A recent paper by Lucas-Serrano et al. indicates that a high-resolution central (HRC) scheme is robust enough to yield accurate hydrodynamical simulations of special relativistic flows in the presence of ultrarelativistic speeds and strong shock waves. In this paper we apply this scheme in full general relativity (involving {\it dynamical} spacetimes), and assess its suitability by performing test simulations for oscillations of rapidly rotating neutron stars and merger of binary neutron stars. It is demonstrated that this HRC scheme can yield results as accurate as those by the so-called high-resolution shock-capturing (HRSC) schemes based upon Riemann solvers. Furthermore, the adopted HRC scheme has increased computational efficiency as it avoids the costly solution of Riemann problems and has practical advantages in the modeling of neutron star spacetimes. Namely, it allows simulations with stiff equations of state by successfully dealing with very low-density unphysical atmospheres. These facts not only suggest that such a HRC scheme may be a desirable tool for hydrodynamical simulations in general relativity, but also open the possibility to perform accurate magnetohydrodynamical simulations in curved dynamic spacetimes.

Assessment of a non-traditional operator split algorithm for simulation of reactive transport

A non-traditional operator split (OS) scheme for the solution of the advection-diffusion-reaction (ADR) equation is proposed. The scheme is implemented with the recently published central scheme [A. Kurganov, E. Tadmor, New high-resolution central schemes for non-linear conservation laws and convection-diffusion equations, J. Comput. Phys. 160 (2000) 241–282] to accurately simulate advection-reaction processes. The governing partial differential equation (PDE) is split into two PDEs, which are solved sequentially within each time step. Unlike traditional methods, the proposed scheme provides a very efficient method to solve the ADR equation for any value of the grid-Péclet number. An analytical mass balance error analysis shows that the proposed non-traditional scheme incurs a splitting error, which behaves differently to the splitting error incurred in traditional OS schemes. Numerical results are presented to illustrate the robustness of the proposed scheme.

Assessment of a high-resolution central scheme for the solution of the relativistic hydrodynamics equations

We assess the suitability of a recent high-resolution central scheme developed by Kurganov & Tadmor (2000) for the solution of the relativistic hydrodynamics equations. The novelty of this approach relies on the absence of Riemann solvers in the solution procedure. The computations we present are performed in one and two spatial dimensions in Minkowski spacetime. Standard numerical experiments such as shock tubes and the relativistic flat-faced step test are performed. As an astrophysical application the article includes two-dimensional simulations of the propagation of relativistic jets using both Cartesian and cylindrical coordinates. The simulations reported clearly show the capabilities of the numerical scheme to yield satisfactory results, with an accuracy comparable to that obtained by the so-called high-resolution shock-capturing schemes based upon Riemann solvers (Godunov-type schemes), even well inside the ultrarelativistic regime. Such central scheme can be straightforwardly applied to hyperbolic systems of conservation laws for which the characteristic structure is not explicitly known, or in cases where the exact solution of the Riemann problem is prohibitively expensive to compute numerically. Finally, we present comparisons with results obtained using various Godunov-type schemes as well as with those obtained using other high-resolution central schemes which have recently been reported in the literature.

Simulation of multicomponent flows using high order central schemes

This paper concerns with the conservative extension of the classical Euler equations to the multicomponent flows. We use high-resolution central schemes to solve these equations. The equilibrium states for each component are coupled in space and time to have a common temperature and velocity. Usually conservative Euler solvers for the gas mixtures produces nonphysical oscillations near contact discontinuities, if the temperature and the ratio of specific heats both are not constant there. However in the current schemes the oscillations near the interfaces are negligible. The schemes also guarantee the exact mass conservation for each component and the exact conservation of total momentum and energy in the whole particle system. The central schemes are robust, reliable, compact and easy to implement. Several one- and two-dimensional numerical test cases are included in this paper, which validates the application of these schemes to multicomponent flows.

Simulation of multicomponent flows using high order central schemes

This paper is concerned with numerical methods for the conservative extension of the classical Euler equations to multicomponent flows. We use high-resolution central schemes to solve these equations. The equilibrium states for each component are coupled in space and time to have a common temperature and velocity. Usually conservative Euler solvers for the gas mixtures produces nonphysical oscillations near contact discontinuities, if the temperature and the ratio of specific heats both are not constant there. However in the schemes considered here the oscillations near the interfaces are negligible. The schemes also guarantee the exact mass conservation for each component and the exact conservation of total momentum and energy in the whole particle system. The central schemes are robust, reliable, compact and easy to implement. Several one- and two-dimensional numerical test cases are included in this paper, which validates the application of these schemes to multicomponent flows.

On gravity and centrifugal settling of polydisperse suspensions forming compressible sediments

A mathematical model describing both the hindered settling and the consolidation of suspensions with particles of different sizes and densities forming compressible sediments is presented. The specific new element is a centrifugal configuration, which gives rise to a non-constant body force. Within a range of angular velocities, the model can be reduced to one (radial) space dimension. The result is a system of second-order strongly degenerate parabolic–hyperbolic convection–diffusion equations. For the special subcase of suspensions of rigid spheres, which do not form compressible sediments and for which the effective solid stress can be assumed to vanish, these equations form a first-order system of conservation laws. A type analysis shows that these equations include hyperbolic, hyperbolic–parabolic, and hyperbolic–elliptic systems, depending on the sizes and densities of the solid particles. A numerical high-resolution central difference scheme for the hyperbolic and hyperbolic–parabolic models is applied to solve the model numerically, and thereby to simulate centrifugation of two polydisperse suspensions.

A Central-Difference Scheme for a Pure Stream Function Formulation of Incompressible Viscous Flow

We present a numerical scheme for incompressible viscous flow, formulated as an equation for the stream function. The pure stream function formulation obviates the difficulty associated with vorticity boundary conditions. The resulting biharmonic equation is discretized with a compact scheme and solved with an algebraic multigrid solver. The advection of vorticity is implemented with a high-resolution central scheme that remains stable and accurate in the presence of large gradients. The accuracy and robustness of the method are demonstrated for high Reynolds number flows in a lid-driven cavity.

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.

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.