Properties of optimal schemes for linear 1D PDE initial-value hyperbolic problems with variable coefficients

This paper concerns a class of problem-dependent schemes referred to as optimal schemes to solve linear PDE initial-value problems with variable coefficients. In this paper both tweo-point and three-point optimal schemes for solutions of a special class of hyperbolic problems have been derived. Results from the study of a sinusoidal and a unit-step wave propagation confirm that, among comparable finite-difference schemes, the optimal scheme produces the solution with minimum truncation error in the L2-norm. This error minimization gives error-free numerical sinusoidal wave propagation. It is suspected that the consistency requirement in the usual derivation of schemes induces errors in the solutions, and it is believed that proper problem modification can lead in general to error-free computation. Numerical data have been generated to confirm our findings.