Strict stability of high-order compact implicit finite-difference schemes:The role of boundary conditions for hyperbolic PDEs, I

Temporal, or "strict" stability of approximation to PDEs is much more diffi cult to achieve than the "classical" Lax stability. In this paper, we prese nt a class of finite-difference schemes for hyperbolic initial boundary val ue problems in one and two space dimensions that possess the property of st rict stability. The approximations are constructed so that all eigenvalues of corresponding differentiation matrix have a nonpositive real part. Bound ary conditions are imposed by using penalty-like terms. Fourth- and sixth-o rder compact implicit finite-difference schemes are constructed and analyze d. Computational efficacy of the approach is corroborated by a series of nu merical tests in 1-D and 2-D scalar problems. (C) 2000 Academic Press.