Reinterpretation of Multi-Stage Methods for Stiff Systems: A Comprehensive Review on Current Perspectives and Recommendations

In this paper, we compare a multi-step method and a multi-stage method for stiff initial value problems. Traditionally, the multi-step method has been preferred than the multi-stage for a stiff problem, to avoid an enormous amount of computational costs required to solve a massive linear system provided by the linearization of a highly stiff system. We investigate the possibility of usage of multi-stage methods for stiff systems by discussing the difference between the two methods in several numerical experiments. Moreover, the advantages of multi-stage methods are heuristically presented even for nonlinear stiff systems through several numerical tests.

DEVELOPMENT OF NEW HARMONIC EULER USING NONSTANDARD FINITE DIFFERENCE TECHNIQUE FOR SOLVING STIFF PROBLEMS

Solving stiff problem always required very tiny size of meshes if it is solved via traditional numerical algorithm. Using insufficient of mesh size, will triggered instabilities. In this paper, we develop an algorithm applying Harmonic Mean on Euler method to solve the stiff problems. The main purpose of this paper is to discuss the improvement of Harmonic Euler using Nonstandard Finite Difference (NSFD). The combination of these methods can provide new advantages that Euler method could offer. Four set of stiff problems are solved via three schemes, i.e. Harmonic Euler, Nonstandard Harmonic Euler and Nonstandard EO with Harmonic Euler. Findings show that both nonstandard schemes produce high accuracy results.

A test of numerical instability and stiffness in the parametrizations of the ARPÉGE and ALADIN models

Meteorological numerical weather prediction (NWP) models solve a system of partial differential equations in time and space. Semi-lagrangian advection schemes allow for long time steps. These longer time steps can result in instabilities occurring in the model physics. A system of differential equations in which some solution components decay more rapidly than others is stiff. In this case it is stability rather than accuracy that restricts the time step. The vertical diffusion parametrization can cause fast non-meteorological oscillations around the slowly evolving true solution (fibrillations). These are treated with an anti-fibrillation scheme, but small oscillations remain in operational weather forecasts using ARPÉGE and ALADIN models. In this paper, a simple test is designed to reveal if the formulation of particular a physical parametrization is a stiff problem or potentially numerically unstable in combination with any other part of the model. When the test is applied to a stable scheme, the solution remains stable. However, applying the test to a potentially unstable scheme yields a solution with fibrillations of substantial amplitude. The parametrizations of the NWP model ARPÉGE were tested one by one to see which one may be the source of unstable model behaviour. The test identified the set of equations in the stratiform precipitation scheme (a diagnostic Kessler-type scheme) as a stiff problem, particularly the combination of terms arising due to the evaporation of snow.

A test of numerical instability and stiffness in the parametrizations of the ARPÉGE and ALADIN models

Meteorological numerical weather prediction (NWP) models solve a system of partial differential equations in time and space. Semi-lagrangian advection scheme in the model dynamics allows for long time-steps. These longer time-steps can result in instabilities occurring in the model physics. A system of differential equations in which some solution components decay more rapidly than others is stiff. In this case it is stability rather than accuracy that restricts the time-step. The vertical diffusion parametrization can cause fast non-meteorological oscillations around the slowly evolving true solution (fibrillations). These are treated with an anti-fibrillation scheme. But small oscillations remain in an operational weather forecasts using ARPÉGE and ALADIN models. It is needed to test of the complete model formulation, as implemented in the operational forecast. In this paper, a simple test is designed. The test reveals if the formulation of particular physical parametrization is a stiff problem or potentially numerically unstable in combination with any other part of the model. When the test is applied to a stable scheme, the solution remains stable. But, applying the test to a potentially unstable scheme yields a solution with fibrillations of substantial amplitude. The parametrizations of a NWP model ARPÉGE were tested one by one to see which one may be the source of unstable model behaviour. The test has identified the stratiform precipitation scheme (a diagnostic Kessler type scheme) as a stiff problem, particularly the term that describes the evaporation of snow.

Homogenization of the Torsion Problem and the Neumann Problem in Nonregular Periodically Perforated Domains

This paper is devoted to the homogenization of the torsion problem (or stiff problem) and the Neumann problem (or soft problem) for second-order elliptic but not necessarily symmetric linear operators set in a bounded open subset Ω of ℝN. More precisely, we study the asymptotic behavior of the equations [Formula: see text] in Ω with ε → 0, where Sε is a closed subset of Ω, which represents the set of the inclusions for the stiff problem or the holes for the soft one, and Ωε = Ω \ Sε. The stiff problem corresponds to δ → + ∞ and ν = 0, the soft one to δ → 0 and ν = 1. We prove a homogenization result in the periodic case without assuming any regularity on the set Sε and thus generalizing the result of Cioranescu and Saint Jean Paulin.7