New implicit schemes for solving a system of conservation laws in one space dimension are obtained by using the cubic-spline technique. By making use of certain perturbation terms, these implicit schemes have been transformed to dissipative schemes. The nonlinear instabilities appearing in the solution in the narrow shock region have been damped by applying the automatic switched Shuman-filter method. Four test examples with continuous and discontinuous initial conditions have been solved to illustrate the theory. The proposed method has been extended to solve a system of conservation laws in two space dimensions.
1. Introduction
Nonlinear hyperbolic systems arise in several areas like gas dynamics, astrophysics and meteorology. In recent years, many f'mite-difference schemes have been proposed for solving these systems of conservation laws (cf. Richtmyer [1], Gourlay and Morris [2], Rubin and Burstein [3], McGuire and Morris [4]). As these schemes are explicit, they involve a severe restriction on the time step. Some implicit schemes (cf. Gourlay and Morris [5], Gary [6], Abarbanel and Zwas [7], Beam and Warming [8]) have also been proposed for solving these systems; these schemes are nondissipative. However, McGuire and Morris [9] have proposed a dissipative implicit finite-difference scheme of second order for solving such a system.
The order of accuracy in all these schemes ranges from order one to order four. Numerical results obtained by the first-order schemes show smooth profiles of discontinuities, resisting nonlinear instability; the shocks are found to be very smeared. The use of second-and higherorder schemes give rise to sharp profiles with large oscillations. These abnormal results are due to the presence of higher-order derivatives near shock-like discontinuities. The schemes of third and fourth order are of little practical value because of their complexity demanding an excessive use of computer time; they also do not provide any specific advantage in handling the discontinuities.
Implicit schemes, in general, are numerically more stable. They are more useful when the instability bound of the explicit scheme is more restrictive than the desired accuracy bound.
In the present paper, we have used the cubic-spline technique and certain perturbation terms for devising a general second-order dissipative implicit scheme.