Implicit CIP (Cubic-Interpolated Propagation) method in one dimension Original Research Article

A new implicit numerical solver for hyperbolic equations, based on the explicit CIP (Cubic-Interpolated Propagation) method, is proposed. Both a physical quantity and its spatial derivative are determined so as to obey the given equation. As the explicit CIP method, this method provides a stable and small diffusion result although it has an implicit form. Most importantly, this method, like other implicit schemes, is stable even in a high-CFL computation. In addition, this scheme can be directly solved by non-iterative procedure because of the two-points connected systems although it has third-order accuracy. The scheme is applied to a one-dimensional shock-tube problem accompanied by a region expanding with quite a high velocity.