Stabilization using filters is intended to model and extract the energy lost to resolved scales due to nonlinearity breaking down resolved scales to unresolved scales. This process is highly nonlinear and yet current models for it use linear filters to select the eddies that will be damped. In this report we consider for the first time nonlinear filters which select eddies for damping (simulating breakdown) based on knowledge of how nonlinearity acts in real flow problems. The particular form of the nonlinear filter allows for easy incorporation of more knowledge into the filter process and its computational complexity is comparable to calculating a linear filter of similar form. We then analyze nonlinear filter based stabilization for the Navier-Stokes equations. We give a precise analysis of the numerical diffusion and error in this process.Mathematics Subject Classification (2010). 65M12, 76D05.
The most effective simulations of the multi-physics coupling of groundwater to surface water must involve employing the best groundwater codes and the best surface water codes. Partitioned methods, which solve the coupled problem by successively solving the sub-physics problems, have recently been studied for the Stokes-Darcy coupling with convergence established over bounded time intervals (with constants growing exponentially in). This report analyzes and tests two such partitioned (non-iterative, domain decomposition) methods for the fully evolutionary Stokes-Darcy problem. Under a modest time step restriction of the form Δ ≤ where = (physical parameters) we prove unconditional asymptotic (over 0 ≤ < ∞) stability of both partitioned methods. From this we derive an optimal error estimate that is uniform in time over 0 ≤ < ∞.
We study the numerical approximation of the solutions of a class of nonlinear reaction-diffusion systems modelling predator-prey interactions, where the local growth of prey is logistic and the predator displays the Holling type II functional response. The fully discrete scheme results from a finite element discretisation in space (with lumped mass) and a semi-implicit discretisation in time. We establish a priori estimates and error bounds for the semi discrete and fully discrete finite element approximations. Numerical results illustrating the theoretical results and spatiotemporal phenomena are presented in one and two space dimensions. The class of problems studied in this paper are real experimental systems where the parameters are associated with real kinetics, expressed in nondimensional form. The theoretical techniques were adapted from a previous study of an idealised reaction-diffusion system (Garvie and Blowey in Eur J Appl Math 16(5):621-646, 2005).
Mathematics Subject Classification (2000)65M60 · 65M15 · 65M12 · 92D25 · 35K55 · 35K57
Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids, involving NavierStokes (NSE) equations in fluid dynamics and Maxwell equations in eletromagnetism. The physical processes of fluid flows and electricity and magnetism are quite different and numerical simulations of each subprocess can require different meshes, time steps, and methods. In most terrestrial applications, MHD flows occur at low-magnetic Reynold numbers. We introduce two partitioned methods to solve evolutionary MHD equations in such cases. The methods we study allow us at each time step to call NSE and Maxwell codes separately, each possibly optimized for the subproblem's respective physics. Complete error analysis and computational tests supporting the theory are given.
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