Stability of explicit advection schemes. The balance point location rule

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AbstractThis paper introduces the balance point location rule, providing specific necessary and sufficient conditions for constructing unconditionally stable explicit advection schemes, in both semi‐Lagrangian and flux‐form Eulerian formulations. The rule determines how the spatial stencil is placed on the computational grid. It requires the balance point (the center of the stencil in index space) to be located in the same patch as the departure point for semi‐Lagrangian schemes or the same cell as the sweep point for Eulerian schemes. Centering the stencil in this way guarantees stability, regardless of the size of the time step. In contrast, the original Courant–Friedrichs–Lewy (CFL) condition requiring the stencil merely to include the departure (sweep) point, although necessary, is not sufficient for guaranteeing stability. The CFL condition is of limited practical value, whereas the balance point location rule always gives precise and easily implemented prescriptions for constructing stable algorithms. The rule is also helpful in correcting a number of misconceptions that have arisen concerning explicit advection schemes. In particular, explicit Eulerian schemes are widely believed to be inefficient because of stability constraints on the time step, dictated by a narrow interpretation of the CFL condition requiring the Courant number to be less than or equal to one. However, such constraints apply only to a particular class of advection schemes resulting for centering the stencil on the arrival point, when in fact the sole function of the stencil is to estimate the departure (sweep) point value—the arrival point has no relevance in determining the placement of the stencil. Unconditionally stable explicit Eulerian advection schemes are efficient and accurate, comparable in operation count to semi‐Lagrangian schemes of the same order, but because of their flux‐based formulation, they have the added advantage of being inherently conservative. Copyright © 2002 John Wiley & Sons, Ltd.