Solving Conservation Laws with Parabolic and Cubic Splines

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Abstract
A continuous curvature cubic-spline method for the treatment of the continuity equation is presented. It is shown that this conservative algorithm, which is second order in time and third order in space, has good amplitude and phase properties. In addition, a positive-definite version is formulated by locally breaking the continuous curvature property. A modified second-order Crowley scheme is also presented, which is positive definite. Results from one- and two-dimensional solid-body rotational-flow experiments, as well as from a two-dimensional deformational flow test, are given. The behavior in a one-dimensional divergent flow is illustrated with solutions of the inviscid Burgers equation. The schemes are extended to dimensions higher than one by Strang's operator-splitting technique. Due to this drawback the schemes are restricted to Courant numbers less than or equal to one.

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