Order Accurate Finite Difference Scheme Research Articles

Flexural waves in beams with geometrical and material longitudinal discontinuities

The flexural wave propagation effects in a linearly elastic Timoshenko beam having both cross-sectional and material type of longitudinal discontinuities. For purposes general formulation is presented which accounts for any number of discontinuities. For purposes of solution the second order accurate explicit MacCormack's finite difference scheme is modified to account for the source term in Timoshenko beam equations and to include the effects of discontinuities. To demonstrate the effectiveness of the formulation and numerical procedure, some representative results are given for a single longitudinal discontinuity in cross-section and in material.

High resolution schemes for hyperbolic conservation laws

A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurate scheme to an appropriately modified flux function. The so-derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme. Numerical experiments are presented to demonstrate the performance of these new schemes.

On a fourth order accurate implicit finite difference scheme for hyperbolic conservation laws. II. Five-point schemes

This paper presents a family of two-level five-point implicit schemes for the solution of one-dimensional systems of hyperbolic conservation laws, which generalized the Crank-Nicholson scheme to fourth order accuracy (4-4) in both time and space. These 4-4 schemes are nondissipative and unconditionally stable. Special attention is given to the system of linear equations associated with these 4-4 implicit schemes. The regularity of this system is analyzed and efficiency of solution-algorithms is examined. A two-datum representation of these 4-4 implicit schemes brings about a compactification of the stencil to three mesh points at each time-level. This compact two-datum representation is particularly useful in deriving boundary treatments. Numerical results are presented to illustrate some properties of the proposed scheme.

On a fourth order accurate implicit finite difference scheme for hyperbolic conservation laws. I. Nonstiff strongly dynamic problems

An implicit finite difference method of fourth order accuracy (in space and time) is introduced for the numerical solution of one-dimensional systems of hyperbolic conservation laws. The basic form of this method is a straightforward generalization of the Crank-Nicholson scheme: it is a two-level scheme which is unconditionally stable and nondissipative. The scheme is compact, i.e., it uses only 3 mesh points at level t and 3 mesh points at level t + Δ t t + \Delta t . In this paper, the first in a series, we present a dissipative version of the basic method which is conditionally stable under the CFL (Courant-Friedrichs-Lewy) condition. This version is particularly useful for numerical solution of problems with strong but nonstiff dynamic features, where the CFL restriction is reasonable on accuracy grounds. Numerical results are presented to illustrate properties of the proposed scheme.

Switched numerical Shuman filters for shock calculations

When using Shuman's filtering operator in the numerical computation of shock waves, nonlinear instabilities are prevented, but high order accuracy is lost even in smooth regions. In order to preserve second or higher order accuracy in these regions, an automatic switched Shuman filter is constructed. Nonsteady shock calculations in one and two spatial dimensions, demonstrate the usefulness and accuracy of the method, including examples with third and fourth order accurate finite difference schemes.