Hybrid-Cubic-Rational Semi-Lagrangian Method with the Optimal Mixing

TL;DR

This work addresses the one-dimensional advection equation \(\frac{\partial f}{\partial t} + u\frac{\partial f}{\partial x} = 0\) using a semi-Lagrangian CIP framework. It introduces a hybrid interpolation that combines cubic and rational components with an optimally determined weight \(\alpha\) to preserve convexity while enhancing accuracy. The authors derive the optimal mixing rule, present a compact expression for the interpolant \(F(k)\) and its derivatives, and validate the method through numerical tests against CIP, conventional rational, and modified rational schemes. The results demonstrate improved accuracy and convexity preservation, with discussion on extensions to multidimensional and conservative formulations.

Abstract

A semi-Lagrangian method for advection equation with hybrid cubic-rational interpolation is introduced. In the present method, the spatial profile of physical quantities is interpolated with a combination of a cubic and a rational function. For achieving both high accuracy and convexity preserving of solution, the two functions are mixed in the optimal ratio which is given theoretically. Accuracy and validity of this method is demonstrated with some numerical experiments.

Figures (9)

• Figure 1: Numerical error as a function of the grid width $h$ in the results with the present, the CIP, the conventional rational and the modified rational (with the additional switching technique) methods.
• Figure 2: Linear propagation of a triangular and a square waves. The figure shows results at $n = 1,000$ (the upper group) and $n = 10,000$ (the lower group) with the present (1), the CIP (2), the conventional rational (3) and the modified rational (4) methods.
• Figure 3: Results of the linear propagation of a square wave at $n = 150$ (the upper group) and $n = 10,000$ (the lower group) with the present (1), the CIP (2), the conventional rational (3) and the modified rational (4) methods. The circles and the solid lines show $f$ and $p$, respectively.
• Figure 4: Results with the reduced mixing ratio (0.99, 0.95, 0.9, 0.8 and 0.7 times values of the optimal one) at $n = 150$ (the upper group) and $n = 10,000$ (the lower group).
• Figure 5: Initial condition of the extreme example. The upper and the lower figures show $f$ and $u$, respectively.