A simple semi‐Lagrangian scheme for advection equations

A semi-Lagrangian scheme is proposed by using forward trajectories and a splitting approach to the interpolation procedure. Several polynomial interpolation schemes are tested, including cubic spline and Lagrange polynomials of degree 3, 5 and 7. A simple filter is also proposed to eliminate spurious short waves and to achieve positive-definiteness. Uniform, rotational and Smolarkiewicz's deformational flows are tested, a solution to the inviscid Burger's equation is also provided. This new algorithm employing cubic-spline interpolation and the new filter yields efficient and accurate short-term simulations. The advantage in efficiency of adopting split trajectories for interpolation is accompanied by a restriction which is slightly more stringent than the stability criterion for conventional semi-Lagrangian advection schemes. It is also found that the simple filter can eliminate spurious short waves very effectively without degrading the solutions.

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