Stability analysis of implicit semi-Lagrangian methods for numerical solution of non-hydrostatic atmospheric dynamics equations
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Vladimir V. Shashkin
Abstract
The stability of implicit semi-Lagrangian schemes for time-integration of the non-hydrostatic atmosphere dynamics equations is analyzed in the present paper. The main reason for the instability of the considered class of schemes is the semi-Lagrangian advection of stratified thermodynamic variables coupled to the fixed point iteration method used to solve the implicit in time upstream trajectory computation problem. We identify two types of unstable modes and obtain stability conditions in terms of the scheme parameters. Stabilization of sound modes requires the use of a pressure reference profile and time off-centering. Gravity waves are stable only for an even number of fixed point method iterations. The maximum time step is determined by inverse buoyancy frequency in the case when the reference profile of the potential temperature is not used. Generally, applying time off-centering and reference profile to pressure variable is necessary for stability. Using reference profile for potential temperature and an even number of the iterations allows one to significantly increase the maximum time-step value.
Acknowledgment
Author is grateful to Mikhail Tolstykh, Gordey Goyman, Rostislav Fadeev, and Nikolay Iakovlev from INM RAS for their comments on the article.
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Funding: The work was carried out at Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences with support of the Russian Science Foundation grant No. 19-71-00160.
A Explicit formula for the Kth approximation in an iterative process
The explicit formula for Kth approximation in the iterative process (3.7) with the initial guess ψ(0) = ψn is written as
where C = R – (I – Q) = (I – αB)–1 Δt(A + B).
One can multiply the sum in equation (A.1) by the identity matrix in the form I = (I – Q)–1(I – Q) and use (I – Q)(I + Q + ... + QK–1) = (I – QK). The result will be
Then, using ψn + (I – Q)–1C = (I – αA – αB)–1(I + βA + βB)ψn = ψn+1 one can obtain the following equality
which is correct independent of the iterative process (3.7) convergence.
One can expand equation (A.3) in the terms of A and B matrices and derive equation (3.9):
where Q͠ = αA(I – βB)–1.
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Artikel in diesem Heft
- Frontmatter
- Comparison of nonlinear solvers within continuation method for steady-state variably saturated groundwater flow modelling
- A regularized isothermal phase-field model of two-phase solid–fluid mixture and its spatial dissipative discretization equations
- New correlative randomized algorithms for statistical modelling of radiation transfer in stochastic medium
- The suite of Taylor–Galerkin class schemes for ice transport on sphere implemented by the INMOST package
- Stability analysis of implicit semi-Lagrangian methods for numerical solution of non-hydrostatic atmospheric dynamics equations
Artikel in diesem Heft
- Frontmatter
- Comparison of nonlinear solvers within continuation method for steady-state variably saturated groundwater flow modelling
- A regularized isothermal phase-field model of two-phase solid–fluid mixture and its spatial dissipative discretization equations
- New correlative randomized algorithms for statistical modelling of radiation transfer in stochastic medium
- The suite of Taylor–Galerkin class schemes for ice transport on sphere implemented by the INMOST package
- Stability analysis of implicit semi-Lagrangian methods for numerical solution of non-hydrostatic atmospheric dynamics equations
