Splitting Methods for Problems with Different Timescales

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  • 1 Cooperative Institute for Research in the Atmosphere, Colorado State University, Foothills Campus, Fort Collins, Colorado
  • | 2 Department of Mathematics, University of California, Los Angeles, Los Angeles, California
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Abstract

The time step for the leapfrog scheme for a symmetric hyperbolic system with multiple timescales is limited by the Courant-Friedlichs-Lewy condition based on the fastest speed present. However, in many physical cases, most of the energy is in the slowest wave, and for this wave the use of the above time step implies that the time truncation error is much smaller than the spatial truncation error. A number of methods have been proposed to overcome this imbalance—for example, the semi-implicit method and the additive splitting technique originally proposed by Marchuk with variations attributable to Strang, and Klemp and Wilhelmson. An analysis of the Marchuk splitting method for multiple timescale systems shows that if a time step based on the slow speed is used, the accuracy of the method cannot be proved, and in practice the method is quite inaccurate. If a time step is chosen that is between the two extremes, then the Klemp and Wilhelmson method can be used, but only if an ad hoc stabilization mechanism is added. The additional computational burden required to maintain the accuracy and the stability of the split-explicit method leads to the conclusion that it is no more efficient than the leapfrog method trivially modified to handle computationally expensive smooth forcing terms.

Using the mathematical analysis developed in a previous manuscript, it is shown that splitting schemes are not appropriate for badly skewed hyperbolic systems. In a number of atmospheric models, the semi-implicit method is used to treat the badly skewed vertical sound wave terms. This leads to the excitation of the high-frequency waves in a nonphysical manner. It is also shown that this is equivalent to solving the primitive equations; that is, a model using this method for the large-scale cast will be ill posed at the lateral boundaries. The multiscale system for meteorology was introduced by Browning and Kreiss to overcome exactly these problems.

Abstract

The time step for the leapfrog scheme for a symmetric hyperbolic system with multiple timescales is limited by the Courant-Friedlichs-Lewy condition based on the fastest speed present. However, in many physical cases, most of the energy is in the slowest wave, and for this wave the use of the above time step implies that the time truncation error is much smaller than the spatial truncation error. A number of methods have been proposed to overcome this imbalance—for example, the semi-implicit method and the additive splitting technique originally proposed by Marchuk with variations attributable to Strang, and Klemp and Wilhelmson. An analysis of the Marchuk splitting method for multiple timescale systems shows that if a time step based on the slow speed is used, the accuracy of the method cannot be proved, and in practice the method is quite inaccurate. If a time step is chosen that is between the two extremes, then the Klemp and Wilhelmson method can be used, but only if an ad hoc stabilization mechanism is added. The additional computational burden required to maintain the accuracy and the stability of the split-explicit method leads to the conclusion that it is no more efficient than the leapfrog method trivially modified to handle computationally expensive smooth forcing terms.

Using the mathematical analysis developed in a previous manuscript, it is shown that splitting schemes are not appropriate for badly skewed hyperbolic systems. In a number of atmospheric models, the semi-implicit method is used to treat the badly skewed vertical sound wave terms. This leads to the excitation of the high-frequency waves in a nonphysical manner. It is also shown that this is equivalent to solving the primitive equations; that is, a model using this method for the large-scale cast will be ill posed at the lateral boundaries. The multiscale system for meteorology was introduced by Browning and Kreiss to overcome exactly these problems.

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