Second‐order accuracy of two‐time‐level semi‐Lagrangian schemes

The accuracy of the two-time-level semi-Lagrangian semi-implicit time integration scheme is analysed. Two relatively independent problems are identified–the discretization of the implicit trajectory equation and the treatment of the nonlinear residual. Two theorems giving conditions for second-order accuracy of the two-time-level semi-Lagrangian semi-implicit scheme are stated. As a result, a new scheme for the trajectory equation is proposed and a class of schemes for the nonlinear residual, which are second-order accurate in time, are introduced. A simplified one-dimensional ‘shallow water’ model is developed in order to test the proposed scheme in comparison with other, previously known, ones. Some experiments proving stability, accuracy and conservation ability are presented.

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