A class of high resolution explicit and implicit shock-capturing methods

An attempt is made to give a unified and generalized formulation of a class of high resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock wave computations. Included is a systematic review of the basic design principle of the various related numerical methods. Special emphasis is on the construction of the basis nonlinear, spatially second and third order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and the flux vector splitting approaches. Generalization of these methods to efficiently include equilibrium real gases and large systems of nonequilibrium flows are discussed. Some issues concerning the applicability of these methods that were designed for homogeneous hyperbolic conservation laws to problems containing stiff source terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for 1-, 2- and 3-dimensional gas dynamics problems

Implicit Total Variation Diminishing (TVD) schemes for steady-state calculations

The application of a new implicit unconditionally stable high resolution total variation diminishing (TVD) scheme to steady state calculations. It is a member of a one parameter family of explicit and implicit second order accurate schemes developed by Harten for the computation of weak solutions of hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a rapid convergence rate, but also generates a highly resolved approximation to the steady state solution. A detailed implementation of the implicit scheme for the one and two dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one and two dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme

Development of a New Class of High order Implicit Non-Oscillatory Schemes for Conservation Laws

Hyperbolic conservation laws allow for discontinuities to develop in the solution. In order to obtain non-oscillatory solutions near discontinuities and high gradient regions, numerical schemes have to satisfy conditions additional to linear stability requirements. These restrictions render high order implicit time integration schemes impractical since the allowable time-step sizes are not much higher than that for explicit schemes. In this work, an investigation is made on the factors that cause these time-step restrictions and two novel schemes are developed in an attempt to overcome the severity of these restrictions. In the first method, the order of accuracy of the {\em time integration} is lowered in high gradient and discontinuous regions. In the second method, the solution is reconstructed in {\em space and time} in a non-oscillatory manner. These concepts are evaluated on model scalar and vector hyperbolic conservation equations. The ultimate objective of this work is to develop a scheme that is accurate and unconditionally non-oscillatory

High Resolution Schemes for Conservation Laws With Source Terms.

This memoir is devoted to the study of the numerical treatment of source terms in hyperbolic conservation laws and systems. In particular, we study two types of situations that are particularly delicate from the point of view of their numerical approximation: The case of balance laws, with the shallow water system as the main example, and the case of hyperbolic equations with stiff source terms. In this work, we concentrate on the theoretical foundations of highresolution total variation diminishing (TVD) schemes for homogeneous scalar conservation laws, firmly established. We analyze the properties of a second order, flux-limited version of the Lax-Wendroff scheme which avoids oscillations around discontinuities, while preserving steady states. When applied to homogeneous conservation laws, TVD schemes prevent an increase in the total variation of the numerical solution, hence guaranteeing the absence of numerically generated oscillations. They are successfully implemented in the form of flux-limiters or slope limiters for scalar conservation laws and systems. Our technique is based on a flux limiting procedure applied only to those terms related to the physical flow derivative/Jacobian. We also extend the technique developed by Chiavassa and Donat to hyperbolic conservation laws with source terms and apply the multilevel technique to the shallow water system. With respect to the numerical treatment of stiff source terms, we take the simple model problem considered by LeVeque and Yee. We study the properties of the numerical solution obtained with different numerical techniques. We are able to identify the delay factor, which is responsible for the anomalous speed of propagation of the numerical solution on coarse grids. The delay is due to the introduction of non equilibrium values through numerical dissipation, and can only be controlled by adequately reducing the spatial resolution of the simulation. Explicit schemes suffer from the same numerical pathology, even after reducing the time step so that the stability requirements imposed by the fastest scales are satisfied. We study the behavior of Implicit-Explicit (IMEX) numerical techniques, as a tool to obtain high resolution simulations that incorporate the stiff source term in an implicit, systematic, manner

Asymptotic Preserving time-discretization of optimal control problems for the Goldstein-Taylor model

We consider the development of implicit-explicit time integration schemes for optimal control problems governed by the Goldstein-Taylor model. In the diffusive scaling this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman-Enskog type limiting procedure. For the class of stiffly accurate implicit-explicit Runge-Kutta methods (IMEX) the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior

On the implementation of a class of upwind schemes for system of hyperbolic conservation laws

The relative computational effort among the spatially five point numerical flux functions of Harten, van Leer, and Osher and Chakravarthy is explored. These three methods typify the design principles most often used in constructing higher than first order upwind total variation diminishing (TVD) schemes. For the scalar case the difference in operation count between any two algorithms may be very small and yet the operation count for their system counterparts might be vastly different. The situation occurs even though one starts with two different yet equivalent representations for the scalar case

Numerical methods for nonlinear Dirac equation

This paper presents a review of the current state-of-the-art of numerical methods for nonlinear Dirac (NLD) equation. Several methods are extendedly proposed for the (1+1)-dimensional NLD equation with the scalar and vector self-interaction and analyzed in the way of the accuracy and the time reversibility as well as the conservation of the discrete charge, energy and linear momentum. Those methods are the Crank-Nicolson (CN) schemes, the linearized CN schemes, the odd-even hopscotch scheme, the leapfrog scheme, a semi-implicit finite difference scheme, and the exponential operator splitting (OS) schemes. The nonlinear subproblems resulted from the OS schemes are analytically solved by fully exploiting the local conservation laws of the NLD equation. The effectiveness of the various numerical methods, with special focus on the error growth and the computational cost, is illustrated on two numerical experiments, compared to two high-order accurate Runge-Kutta discontinuous Galerkin methods. Theoretical and numerical comparisons show that the high-order accurate OS schemes may compete well with other numerical schemes discussed here in terms of the accuracy and the efficiency. A fourth-order accurate OS scheme is further applied to investigating the interaction dynamics of the NLD solitary waves under the scalar and vector self-interaction. The results show that the interaction dynamics of two NLD solitary waves depend on the exponent power of the self-interaction in the NLD equation; collapse happens after collision of two equal one-humped NLD solitary waves under the cubic vector self-interaction in contrast to no collapse scattering for corresponding quadric case.Comment: 39 pages, 13 figure