Schemes For Hyperbolic Systems Research Articles

Uniform accuracy of implicit-explicit Runge-Kutta (IMEX-RK) schemes for hyperbolic systems with relaxation

Implicit-explicit Runge-Kutta (IMEX-RK) schemes are popular methods to treat multiscale equations that contain a stiff part and a non-stiff part, where the stiff part is characterized by a small parameter ε \varepsilon . In this work, we prove rigorously the uniform stability and uniform accuracy of a class of IMEX-RK schemes for a linear hyperbolic system with stiff relaxation. The result we obtain is optimal in the sense that it holds regardless of the value of ε \varepsilon and the order of accuracy is the same as the design order of the original scheme, i.e., there is no order reduction.

A new class of entropy stable schemes for hyperbolic systems: Finite element methods

A new class of entropy stable schemes for hyperbolic systems: Finite element methods

Simulation of THz Oscillations in Semiconductor Devices Based on Balance Equations

Instabilities of electron plasma waves in high-mobility semiconductor devices have recently attracted a lot of attention as a possible candidate for closing the THz gap. Conventional moments-based transport models usually neglect time derivatives in the constitutive equations for vectorial quantities, resulting in parabolic systems of partial differential equations (PDE). To describe plasma waves however, such time derivatives need to be included, resulting in hyperbolic rather than parabolic systems of PDEs; thus the fundamental nature of these equation systems is changed completely. Additional nonlinear terms render the existing numerical stabilization methods for semiconductor simulation practically useless. On the other hand there are plenty of numerical methods for hyperbolic systems of PDEs in the form of conservation laws. Standard numerical schemes for conservation laws, however, are often either incapable of correctly handling the large source terms present in semiconductor devices due to built-in electric fields, or rely heavily on variable transformations which are specific to the equation system at hand (e.g. the shallow water equations), and can not be generalized easily to different equations. In this paper we develop a novel well-balanced numerical scheme for hyperbolic systems of PDEs with source terms and apply it to a simple yet non-linear electron transport model.

On stability of the difference splitting scheme for hyperbolic systems

On stability of the difference splitting scheme for hyperbolic systems

Implicit-Explicit Runge--Kutta Schemes for Hyperbolic Systems and Kinetic Equations in the Diffusion Limit

We consider implicit-explicit (IMEX) Runge--Kutta (R-K) schemes for hyperbolic systems with stiff relaxation in the so-called diffusion limit. In such a regime the system relaxes towards a convection-diffusion equation. The first objective of this paper is to show that traditional partitioned IMEX R-K schemes will relax to an explicit scheme for the limit equation with no need of modification of the original system. Of course the explicit scheme obtained in the limit suffers from the classical parabolic stability restriction on the time step. The main goal of this paper is to present an approach, based on IMEX R-K schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the convection-diffusion equation, in which the diffusion is treated implicitly. This is achieved by a novel reformulation of the problem, and subsequent application of IMEX R-K schemes to it. An analysis of such schemes to the reformulated problem shows that the schemes reduce to IMEX R-K schemes for the limit equation, under the same conditions derived for hyperbolic relaxation [S. Boscarino and G. Russo, SIAM J. Sci. Comput., 31 (2009), pp. 1926--1945]. Several numerical examples including neutron transport equations confirm the theoretical analysis.

A Finite Volume Upwind-Biased Centred Scheme for Hyperbolic Systems of Conservation Laws: Application to Shallow Water Equations

AbstractWe construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form. It applies in multidimensional structured and unstructured meshes. The proposed method is an extension of the UFORCE method developed by Stecca, Siviglia and Toro, in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan. The proposed first-order method is shown to be identical to the Godunov upwind method in applications to a 2 x 2 linear hyperbolic system. The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations. Extension to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured meshes. Finally, numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.

Linearly Implicit Approximations of Diffusive Relaxation Systems

Diffusive relaxation systems provide a general framework to approximate nonlinear diffusion problems, also in the degenerate case (Aregba-Driollet et al. in Math. Comput. 73(245):63---94, 2004; Boscarino et al. in Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, 2011; Cavalli et al. in SIAM J. Sci. Comput. 34:A137---A160, 2012; SIAM J. Numer. Anal. 45(5):2098---2119, 2007; Naldi and Pareschi in SIAM J. Numer. Anal. 37:1246---1270, 2000; Naldi et al. in Surveys Math. Indust. 10(4):315---343, 2002). Their discretization is usually obtained by explicit schemes in time coupled with a suitable method in space, which inherits the standard stability parabolic constraint. In this paper we combine the effectiveness of the relaxation systems with the computational efficiency and robustness of the implicit approximations, avoiding the need to resolve nonlinear problems and avoiding stability constraints on time step. In particular we consider an implicit scheme for the whole relaxation system except for the nonlinear source term, which is treated though a suitable linearization technique. We give some theoretical stability results in a particular case of linearization and we provide insight on the general case. Several numerical simulations confirm the theoretical results and give evidence of the stability and convergence also in the case of nonlinear degenerate diffusion.

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.

On a Class of Uniformly Accurate IMEX Runge–Kutta Schemes and Applications to Hyperbolic Systems with Relaxation

In this paper we consider hyperbolic systems with relaxation in which the relaxation time $\varepsilon$ may vary from values of order one to very small values. When $\varepsilon$ is very small, the relaxation term becomes very strong and highly stiff, and underresolved numerical schemes may produce spurious results. In such cases it is important to have schemes that work uniformly with respect to $\varepsilon$. IMplicit-EXplicit (IMEX) Runge–Kutta (R-K) schemes have been widely used for the time evolution of hyperbolic partial differential equations but the schemes existing in literature do not exhibit uniform accuracy with respect to the relaxation time. We develop new IMEX R-K schemes for hyperbolic systems with relaxation that present better uniform accuracy than the ones existing in the literature and in particular produce good behavior with high order accuracy in the asymptotic limit, i.e., when $\varepsilon$ is very small. These schemes are obtained by imposing new additional order conditions to guarantee better accuracy over a wide range of the relaxation time. We propose the construction of new third-order IMEX R-K schemes of type CK [S. Boscarino, SIAM J. Numer. Anal., 45 (2008), pp. 1600–1621]. In several test problems, these schemes, with a fixed spatial discretization, exhibit for all range of the relaxation time an almost uniform third-order accuracy.

A Runge–Kutta, Taylor–Galerkin scheme for hyperbolic systems with source terms. Application to shock wave propagation in viscoplastic geomaterials

This paper presents an alternative formulation of Solid Dynamics problems based on (i) a mathematical model consisting of a system of hyperbolic PDEs where the source term is originated by the viscoplastic strain rate and (ii) a splitting scheme where the two-step Taylor–Galerkin is used for the advective part of the PDE operator while the sources are integrated using a fourth-order Runge–Kutta. Use of the splitting scheme results in a higher accuracy than that of the original two-step Taylor–Galerkin. The scheme performs well when used with linear triangle or tetrahedra for (i) bending-dominated situations (ii) localized failure under dynamic conditions and keeps the advantages of the two-step Taylor–Galerkin concerning numerical dispersion and damping of short wavelengths. Copyright © 2006 John Wiley & Sons, Ltd.

Improved Applications of Relaxation Schemes for Hyperbolic Systems of Conservation Laws and Convection-diffusion Problems

AbstractA class of third-order relaxation schemes for hyperbolic systems of conservation laws with source terms is reconstructed. The schemes employ general higher-order reconstruction for spatial discretization and higher-order implicit-explicit schemes or TVD Runge — Kutta schemes for time integration of relaxed systems. Extension to multidimensional convection-diffusion problems is also included in this paper. Numerical results for inviscid gas Euler equations, shallow water and incompressible Navier — Stokes equations are given in both one and two space dimensions.

Stability and Convergence of a Class of Finite Element Schemes for Hyperbolic Systems of Conservation Laws

We propose a class of finite element schemes for systems of hyperbolic conservation laws that are based on finite element discretizations of appropriate relaxation models. We consider both semidiscrete and fully discrete finite element schemes and show that the schemes are stable and, when the compensated compactness theory is applicable, do converge to a weak solution of the hyperbolic system. The schemes use piecewise polynomials of arbitrary degree and their consistency error is of high order. We also prove that the rate of convergence of the relaxation system to a smooth solution of the conservation laws is of order $O(\eps )$.

A fixed-stencil non-oscillatory scheme for hyperbolic systems

A fixed-stencil non-oscillatory scheme for hyperbolic systems

A Parallel Adaptive Numerical Scheme for Hyperbolic Systems of Conservation Laws

We generalize the first author’s adaptive numerical scheme for scalar first order conservation laws to systems of equations in one space dimension. The resulting numerical method generates highly nonuniform, time-dependent grids, and hence is difficult to execute efficiently on vector computers such as the Cray or Cyber 205. In contrast, we show that this type of algorithm may be executed in parallel on alternate computer architectures. We describe a parallel implementation of the algorithm on the Denelcor HEP, a multiple-instruction, multiple-data (MIMD) shared memory parallel computer. This program dynamically partitions the problem domain so as to achieve effective load balancing among the available processes.

Justification of an averaging scheme for hyperbolic systems with fast and slow variables. The mixed problem

Justification of an averaging scheme for hyperbolic systems with fast and slow variables. The mixed problem

THE APPARENT NUMERICAL STABILITY OF UNPHYSICAL SHOCKS

Difference schemes for hyperbolic systems of conservation laws occasionally converge to an unphysical weak solution, i.e. a weak solution containing discontinuities for which the entropy condition is violated. These unphysical discontinuities, when they exist as solutions of the difference scheme, tend to exhibit a surprising stability under perturbations. We point out here that this can be explained by an energy inequality, which is valid for the discrete approximation but which is not valid as applied to the differential equation itself. In spite of this difficulty, many simple difference schemes are highly successful at converging to the physically correct weak solution. A mechanism for this is given; we show that for shocks of moderate strength, there are numerous quadratic forms in the dependent variables which can serve effectively as entropy functions, i.e. for which an inequality exists determining the physically correct weak solution. It is shown how the limits of the solutions of a difference scheme will often necessarily satisfy such an inequality; as they are generally weak solutions of the given system, they must thus be the correct weak solutions.

A monotonic second-order difference scheme for hyperbolic systems with two independent variables

Considering the example of the equations of one-dimensional non-stationary gas dynamics, a modification of Godunov's well-know scheme is proposed for hyperbolic systems with two independent variables; the modification, while preserving the monotonicity, raises to second order the approximation of the differential operator and reduces smearing of contact discontinuities and low-intensity jumps.

The first differential approximation to finite-difference schemes for hyperbolic systems of equations

The first differential approximation to finite-difference schemes for hyperbolic systems of equations

The relation between the correctness of the first differential approximations and the stability of difference schemes for hyperbolic systems of equations

The concept of the first differential approximation of a difference scheme is introduced and the conditions relating the stability of difference schemes and the correctness of their first differential approximations are established.

Two-Level Difference Schemes for Hyperbolic Systems

Two-Level Difference Schemes for Hyperbolic Systems