Difference Schemes For Hyperbolic Systems Research Articles

On the strong stability of finite difference schemes for hyperbolic systems in two space dimensions

We study the stability of some finite difference schemes for symmetric hyperbolic systems in two space dimensions. For the so-called upwind scheme and the Lax–Wendroff scheme with a stabilizer, we show that stability is equivalent to strong stability, meaning that both schemes are either unstable or $$\ell ^2$$ -decreasing. These results improve on a series of partial results on strong stability. We also show that, for the Lax–Wendroff scheme without stabilizer, strong stability may not occur no matter how small the CFL parameters are chosen. This partially invalidates some of Turkel’s conjectures in Turkel (16(2):109–129, 1977).

On Stability, Monotonicity, and Construction of Difference Schemes II: Applications

In this paper, we focus on the application and illustration of the approach developed in part I. This approach is found to be useful in the construction of stable and monotone central difference schemes for hyperbolic systems. A new modification of the central Lax–Friedrichs scheme is developed to be of second-order accuracy. The stability of several versions of the developed central scheme is proved. Necessary conditions for the variational monotonicity of the scheme are found. A monotone piecewise cubic interpolation is used in the central schemes to give an accurate approximation for the model in question. The monotonicity parameter introduced in part I is found to be important. That parameter, along with the Courant–Friedrichs–Lewy (CFL) number, plays a major role in the criteria for monotonicity and stability of the central schemes. The modified scheme is tested on several conservation laws taking a CFL number equal or close to unity, and the scheme is found to be accurate and robust.

Efficient Characteristic Projection in Upwind Difference Schemes for Hyperbolic Systems: The Complementary Projection Method

The standard construction of upwind difference schemes for hyperbolic systems of conservation laws requires the full eigensystem of the Jacobian matrix. This system is used to define the transformation into and out of the characteristic scalar fields, where upwind differencing is meaningful. When the Jacobian has a repeated eigenvalue, the associated normalized eigenvectors are not uniquely determined, and an arbitrary choice of eigenvectors must be made to span the characteristic subspace. In this report we point out that it is possible to avoid this arbitrary choice entirely. Instead, a complementary projection technique can be used to formulate upwind differencing without specifying a basis. For systems with eigenvalues of high multiplicity, this approach simplifies the analytical and programming effort and reduces the computational cost. Numerical experiments show no significant difference in computed results between this formulation and the traditional one, and thus we recommend its use for these types of problems. This complementary projection method has other applications. For example, it can be used to extend upwind schemes to some weakly hyperbolic systems. These lack complete eigensystems, so the traditional form of characteristic upwinding is not possible.

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.

Upwind Difference Schemes for Hyperbolic Systems of Conservation Laws

Upwind Difference Schemes for Hyperbolic Systems of Conservation Laws

Upwind difference schemes for hyperbolic systems of conservation laws

We derive a new upwind finite difference approximation to systems of nonlinear hyperbolic conservation laws. The scheme has desirable properties for shock calculations. Under fairly general hypotheses we prove that limit solutions satisfy the entropy condition and that discrete steady shocks exist which are unique and sharp. Numerical examples involving the Euler and Lagrange equations of compressible gas dynamics in one and two space dimensions are given.

Accuracy of Third-Order Predictor-Corrector Difference Schemes for Hyperbolic Systems

Accuracy of Third-Order Predictor-Corrector Difference Schemes for Hyperbolic Systems

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

Width of the zone of influence for the equation of heat conduction

A zone of influence, determined by the characteristics, is well known to exist for partial differential equations of the hyperbolic type. There is a close connection between the width of this zone and the stability condition derived when considering explicit difference schemes for hyperbolic systems.

On certain finite difference schemes for hyperbolic systems

schemes for the numerical solution of hyperbolic systems of partial differential equations. Our main concern will be the stability conditions for these schemes although some of these schemes may be useful, especially for problems involving supersonic fluid flow. We will describe numerical experiments designed to check the derived stability conditions and also the accuracy of these schemes. In Section 2 we will describe the application of the Crank-Nicholson scheme to hyperbolic systems. This method is unconditionally stable, however it requires the inversion of a block tridiagonal matrix. We describe two modifications which require only the inversion of a scalar tridiagonal matrix. We carry out a stability analysis which shows that these schemes are unconditionally stable only if the matrix of the system is positive definite (supersonic flow in the case of hydrodynamics). Results of numerical computations using all these schemes are described in Section 3. All our results are for problems in one space dimension. These methods can be generalized to two dimensions but we have no analysis to indicate that the generalizations will work. In Section 5 we give the results of fluid dynamics computations using three versions of the Lax-Wendroff difference scheme. The objective here is to determine if it is necessary to write the equations in conservation form in order to obtain good results when the flow contains a shock. In Section 6 an application of the Lax-Wendroff scheme to the Navier-Stokes equations is described. An empirical stability condition is obtained which is a combination of the usual hyperbolic and parabolic conditions. In Section 7 an explicit difference scheme related to the CrankNicholson scheme is given. This is an iterative scheme with a rather peculiar stability condition. The author is indebted to several staff members at the Couraint Institute of Mathematical Sciences for advice and encouragement, especially Professors R. D. Richtmyer and H. B. Keller. The computations described herein were all carried out on the IBM 7090 at New York University.