Semi-implicit second order schemes for numerical solution of level set advection equation on Cartesian grids

Abstract A new parametric class of semi-implicit numerical schemes for a level set advection equation on Cartesian grids is derived and analyzed. An accuracy and a stability study is provided for a linear advection equation with a variable velocity using partial Lax–Wendroff procedure and numerical von Neumann stability analysis. The obtained semi-implicit κ -scheme is 2nd order accurate in space and time in any dimensional case when using a dimension by dimension extension of the one-dimensional scheme that is not the case for analogous fully explicit or fully implicit κ -schemes. A further improvement is obtained by using so-called Corner Transport Upwind extension in two-dimensional case. The extended semi-implicit κ -scheme with a specific (velocity dependent) value of κ is 3rd order accurate in space and time for a constant advection velocity, and it is unconditional stable according to the numerical von Neumann stability analysis for the linear advection equation in general.

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