Semi-discrete Central Scheme Research Articles

A stiffly stable semi-discrete scheme for the characteristic linear hyperbolic relaxation with boundary

We study the stability of the semi-discrete central scheme for the linear damped wave equation with boundary. We exhibit a sufficient condition on the boundary to guarantee the uniform stability of the initial boundary value problem for the relaxation system independently of the stiffness of the source term and of the space step. The boundary is approximated using a summation-by-parts method and the stiff stability is proved using energy estimates and the Laplace transform. We also investigate if the condition is also necessary, following the continuous case studied by Xin and Xu (J. Differ. Equ. 167 (2000) 388–437).

A semi-discrete central scheme for incompressible multiphase flow in porous media in several space dimensions

In this work we present a Godunov-type semi-discrete central scheme for systems of conservation laws which allows for spatial heterogeneity of the storage coefficient, say, the porosity field. This scheme is used in the composition of a sequential splitting algorithm for simulating incompressible multiphase flow within rigid porous media, with both permeability and porosity fields heterogeneous. The proposed methodology composes a fundamental block in the simulation of complex flows in porous media, where the compressibility effects may be included. Numerical tests are presented to illustrate the accuracy of the proposed method in problems that simulate immiscible three-phase flow in heterogeneous porous media in two and three space dimensions.

A numerical study of non-collinear wave mixing and generated resonant components

Interaction of two non-collinear nonlinear ultrasonic waves in an elastic half-space with quadratic nonlinearity is investigated in this paper. A hyperbolic system of conservation laws is applied here and a semi-discrete central scheme is used to solve the numerical problem. The numerical results validate that the model can be used as an effective method to generate and evaluate a resonant wave when two primary waves mix together under certain resonant conditions. Features of the resonant wave are analyzed both in the time and frequency domains, and variation trends of the resonant waves together with second harmonics along the propagation path are analyzed. Applied with the pulse-inversion technique, components of resonant waves and second harmonics can be independently extracted and observed without distinguishing times of flight. The results show that under the circumstance of non-collinear wave mixing, both sum and difference resonant components can be clearly obtained especially in the tangential direction of their propagation. For several rays of observation points around the interaction zone, the further it is away from the excitation sources, generally the earlier the maximum of amplitude arises. From the parametric analysis of the phased array, it is found that both the length of array and the density of element have impact on the maximum of amplitude of the resonant waves. The spatial distribution of resonant waves will provide necessary information for the related experiments.

A Nonlinear Ultrasound Method for Fatigue Evaluation of Marine Structures

Fatigue is a major cause of failure in marine structures resulting from random wave and wind loading. A nonlinear ultrasound method for fatigue evaluation which uses interaction of two non-collinear nonlinear ultrasonic waves with quadratic nonlinearity is investigated in this paper. A hyperbolic system of conservation laws is applied here and a semi-discrete central scheme is used to solve the numerical problem. The numerical results prove that a resonant wave can be generated by two primary waves with certain resonant conditions. Features of the resonant wave are analyzed both in the time and frequency domains, and several regularities are found on intensity distribution of the resonant wave in two-dimensional domain.

A positivity preserving central scheme for shallow water flows in channels with wet-dry states

We present a high-resolution, non-oscillatory semi-discrete central scheme for one- dimensional shallow-water flows along channels with non uniform cross sections of arbitrary shape and bottom topography. The proposed scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and enjoys two properties crucial for the accurate simulation of shallow- water flows: it preserves the positivity of the water height, and it is well balanced, i.e., the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. In addition to these, a modification in the numerical flux and the estimate of the speed of propagation, the scheme incorporates the ability to detect and resolve partially wet regions, i.e., wet-dry states. Along with a detailed description of the scheme and proofs of its properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.

On the Full and Global Accuracy of a Compact Third Order WENO Scheme

Recently, Aràndiga et al. showed in [SIAM J. Numer. Anal., 49 (2011), pp. 893--915] for a class of weighted ENO (WENO) schemes that the parameter $\varepsilon$ occurring in the smoothness indicators of the scheme should be chosen proportional to the square of the mesh size, $h^2$, to achieve the optimal order of accuracy. Unfortunately, these results cannot be applied to the compact third order WENO reconstruction procedure introduced in [D. Levy, G. Puppo, and G. Russo, SIAM J. Sci. Comput., 22 (2000), pp. 656--672], which we apply within the semidiscrete central scheme of [A. Kurganov and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461--1488], a commonly used scheme for the numerical solution of conservation laws and convection-diffusion equations. The aim of this paper is to close this gap. In particular, we will show that we achieve the optimal order of accuracy in the WENO reconstruction ($h^3$ in the smooth case and $h^2$ near discontinuities) for $\varepsilon = K h^q$ with $q \le 3$ and $pq \ge 2$, where $p \ge 1$ is the exponent used in the computation of the weights in the WENO scheme. Numerical examples showing the predicted order of convergence of the analyzed WENO reconstruction procedure are given as well as results for the presented semidiscrete scheme combined with a third order TVD--Runge--Kutta scheme from [S. Gottlieb and C.-W. Shu, Math. Comp., 67 (1998), pp. 73--85] for the time integration.

A numerical study of non-collinear mixing of three-dimensional nonlinear waves in an elastic half-space

Interactions of two non-collinear nonlinear ultrasonic waves in an elastic half-space with quadratic and hysteretic nonlinearity are investigated in this paper. The numerical problem is solved by a semi-discrete central scheme as well as a finite element method in two and three dimensions, respectively. Features and intensity distribution of the resonant wave are analyzed both in time and frequency domains, and the method of non-collinear wave mixing is proved to be a promising method, which is both effective and sensitive in material characterization and structure damage detection.

A TREATMENT OF CONTACT DISCONTINUITY FOR CENTRAL UPWIND SCHEME BY CHANGING FLUX FUNCTIONS

Central schemes offer a simple and versatile approach for computing approximate solutions of nonlinear systems of hyperbolic conservation laws. However, there are large numerical dissipation in case of contact discontinuity. We study semi-discrete central upwind scheme by changing flux functions to reduce the numerical dissipation and we perform numerical computations for various problems in case of contact discontinuity.

Nonoscillatory Central Schemes for Hyperbolic Systems of Conservation Laws in Three-Space Dimensions

We extend a family of high-resolution, semidiscrete central schemes for hyperbolic systems of conservation laws to three-space dimensions. Details of the schemes, their implementation, and properties are presented together with results from several prototypical applications of hyperbolic conservation laws including a nonlinear scalar equation, the Euler equations of gas dynamics, and the ideal magnetohydrodynamic equations. Parallel scaling analysis and grid-independent results including contours and isosurfaces of density and velocity and magnetic field vectors are shown in this study, confirming the ability of these types of solvers to approximate the solutions of hyperbolic equations efficiently and accurately.

A Non-oscillatory Central Scheme for One-Dimensional Two-Layer Shallow Water Flows along Channels with Varying Width

We present a new high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional two-layer shallow-water flows along channels with non-uniform rectangular cross sections and bottom topography. The scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and it enjoys two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water height, and it is well balanced, i.e., the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. Along with a detailed description of the scheme and proofs of these two properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.

The role of the electron convection term for the parallel electric field and electron acceleration in MHD simulations

There has been a great concern about the origin of the parallel electric field in the frame of fluid equations in the auroral acceleration region. This paper proposes a new method to simulate magnetohydrodynamic (MHD) equations that include the electron convection term and shows its efficiency with simulation results in one dimension. We apply a third-order semi-discrete central scheme to investigate the characteristics of the electron convection term including its nonlinearity. At a steady state discontinuity, the sum of the ion and electron convection terms balances with the ion pressure gradient. We find that the electron convection term works like the gradient of the negative pressure and reduces the ion sound speed or amplifies the sound mode when parallel current flows. The electron convection term enables us to describe a situation in which a parallel electric field and parallel electron acceleration coexist, which is impossible for ideal or resistive MHD.

CENTRAL SCHEMES WITH LAX-WENDROFF TYPE TIME DISCRETIZATIONS

The semi-discrete central scheme and central upwind scheme use Runge-Kutta (RK) time discretization. We do the Lax-Wendroff (LW) type time discretization for both schemes. We perform numerical experiments for various problems including two dimensional Riemann problems for Burgers' equation and Euler equations. The results show that the LW time discretization is more efficient in CPU time than the RK time discretization while maintaining the same order of accuracy.

A semi-discrete central scheme for the approximation of two-phase flows in three space dimensions

Abstract We present a new second-order (in space), semi-discrete, central scheme for the approximation of hyperbolic conservation laws in three space dimensions. The proposed scheme is applied to a model for two-phase, immiscible and incompressible displacement in heterogeneous porous media. Numerical simulations are presented to demonstrate its ability to approximate solutions of hyperbolic equations efficiently and accurately in petroleum reservoir simulations.

A semi-discrete central scheme for magnetohydrodynamics on orthogonal–curvilinear grids

This work describes a novel scheme for the equations of magnetohydrodynamics on orthogonal–curvilinear grids within a finite-volume framework. The scheme is based on a combination of central-upwind techniques for hyperbolic conservation laws and projection–evolution methods originally developed for Hamilton–Jacobi equations. The scheme is derived in semi-discrete form, and a full-fledged version is obtained by applying any stable and accurate solver for integration in time. The divergence-free condition of the magnetic field is a built-in property of the scheme by virtue of a constrained-transport ansatz for the induction equation. From the general formulation second-order accurate schemes for cylindrical grids and spherical grids are introduced in some more detail pointing out their potential importance in many applications. Special emphasis in this context is put to a treatment of the geometric axis implying severe complications because of the presence of coordinate singularities and associated grid degeneracy. An attempt to tackle these problems is presented. Numerical experiments illustrate the overall robustness and performance of the scheme for a small suite of tests.

On the development of a high-performance tool for the simulation of CO2 injection into deep saline aquifers

We report on the development of a multiscale parallel simulator for porous media flow problems. We combine state-of-the-art numerical techniques in a new object-oriented, high-performance simulation tool. The new multiscale parallel software will adapt itself to the type and number of available processing cores. The combination of: physically based operator splitting for multiscale time discretization of nonlinear systems of partial differential equations arising in multiphase flows in porous media, domain decomposition for the parallel solution of elliptic and parabolic problems, and semi-discrete central finite volume schemes for hyperbolic systems allows us to produce new very accurate simulations of multiphase flow in porous media problems that are of interest in many areas of science and technology, such as petroleum reservoir and environmental engineering. The new simulation code may aid the assessment and monitoring of CO 2 sequestration projects by providing accurate predictions of the migration and trapping of injected CO 2 plumes.

Two-dimensional central-upwind schemes for curvilinear grids and application to gas dynamics with angular momentum

In this work we present new second order semi-discrete central schemes for systems of hyperbolic conservation laws on curvilinear grids. Our methods generalise the two-dimensional central-upwind schemes developed by Kurganov and Tadmor [A. Kurganov, E. Tadmor, Numer. Methods Partial Differential Equations 18 (2002) 584. [1]]. In these schemes we account for area and volume changes in the numerical flux functions due to the non-cartesian geometries. In case of vectorial conservation laws we introduce a general prescription of the geometrical source terms valid for various orthogonal curvilinear coordinate systems. The methods are applied to the two-dimensional Euler equations of inviscid gas dynamics with and without angular momentum transport. In the latter case we introduce a new test problem to examine the detailed conservation of specific angular momentum.

Two-dimensional wave propagation in an elastic half-space with quadratic nonlinearity: A numerical study

This study investigates two-dimensional wave propagation in an elastic half-space with quadratic nonlinearity. The problem is formulated as a hyperbolic system of conservation laws, which is solved numerically using a semi-discrete central scheme. These numerical results are then analyzed in the frequency domain to interpret the nonlinear effects, specifically the excitation of higher-order harmonics. To quantify and compare the nonlinearity of different materials, a new parameter is introduced, which is similar to the acoustic nonlinearity parameter beta for one-dimensional longitudinal waves. By using this new parameter, it is found that the nonlinear effects of a material depend on the point of observation in the half-space, both the angle and the distance to the excitation source. Furthermore it is illustrated that the third-order elastic constants have a linear effect on the acoustic nonlinearity of a material.

Central schemes for porous media flows

We are concerned with central differencing schemes for solving scalar hyperbolic conservation laws arising in the simulation of multiphase flows in heterogeneous porous media. We compare the Kurganov-Tadmor, 2000 semi-discrete central scheme with the Nessyahu-Tadmor, 1990 central scheme. The KT scheme uses more precise information about the local speeds of propagation together with integration over nonuniform control volumes, which contain the Riemann fans. These methods can accurately resolve sharp fronts in the fluid saturations without introducing spurious oscillations or excessive numerical diffusion. We first discuss the coupling of these methods with velocity fields approximated by mixed finite elements. Then, numerical simulations are presented for two-phase, two-dimensional flow problems in multi-scale heterogeneous petroleum reservoirs. We find the KT scheme to be considerably less diffusive, particularly in the presence of high permeability flow channels, which lead to strong restrictions on the time step selection; however, the KT scheme may produce incorrect boundary behavior.

Central schemes for porous media flows

We are concerned with central differencing schemes for solving scalar hyperbolic conservation laws arising in the simulation of multiphase flows in heterogeneous porous media. We compare the Kurganov-Tadmor, 2000 semi-discrete central scheme with the Nessyahu-Tadmor, 1990 central scheme. The KT scheme uses more precise information about the local speeds of propagation together with integration over nonuniform control volumes, which contain the Riemann fans. These methods can accurately resolve sharp fronts in the fluid saturations without introducing spurious oscillations or excessive numerical diffusion. We first discuss the coupling of these methods with velocity fields approximated by mixed finite elements. Then, numerical simulations are presented for two-phase, two-dimensional flow problems in multi-scale heterogeneous petroleum reservoirs. We find the KT scheme to be considerably less diffusive, particularly in the presence of high permeability flow channels, which lead to strong restrictions on the time step selection; however, the KT scheme may produce incorrect boundary behavior.

A central scheme for shallow water flows along channels with irregular geometry

We present a new semi-discrete central scheme for one-dimensional shallow water flows along channels with non-uniform rectangular cross sections and bottom topography. The scheme preserves the positivity of the water height, and it is preserves steady-states of rest (i.e., it is well-balanced). Along with a detailed description of the scheme, numerous numerical examples are presented for unsteady and steady flows. Comparison with exact solutions illustrate the accuracy and robustness of the numerical algorithm.