1,774 research outputs found
Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions
The (asymptotic) behaviour of the second moment of solutions to stochastic
differential equations is treated in mean-square stability analysis. This
property is discussed for approximations of infinite-dimensional stochastic
differential equations and necessary and sufficient conditions ensuring
mean-square stability are given. They are applied to typical discretization
schemes such as combinations of spectral Galerkin, finite element,
Euler-Maruyama, Milstein, Crank-Nicolson, and forward and backward Euler
methods. Furthermore, results on the relation to stability properties of
corresponding analytical solutions are provided. Simulations of the stochastic
heat equation illustrate the theory.Comment: 22 pages, 4 figures; deleted a section; shortened the presentation of
results; corrected typo
A Discontinuous Galerkin Method for Ideal Two-Fluid Plasma Equations
A discontinuous Galerkin method for the ideal 5 moment two-fluid plasma
system is presented. The method uses a second or third order discontinuous
Galerkin spatial discretization and a third order TVD Runge-Kutta time stepping
scheme. The method is benchmarked against an analytic solution of a dispersive
electron acoustic square pulse as well as the two-fluid electromagnetic shock
and existing numerical solutions to the GEM challenge magnetic reconnection
problem. The algorithm can be generalized to arbitrary geometries and three
dimensions. An approach to maintaining small gauge errors based on error
propagation is suggested.Comment: 40 pages, 18 figures
The semi-discrete Galerkin finite element modelling of compressible viscous flow past an airfoil
A method is developed to solve the two-dimensional, steady, compressible, turbulent boundary-layer equations and is coupled to an existing Euler solver for attached transonic airfoil analysis problems. The boundary-layer formulation utilizes the semi-discrete Galerkin (SDG) method to model the spatial variable normal to the surface with linear finite elements and the time-like variable with finite differences. A Dorodnitsyn transformed system of equations is used to bound the infinite spatial domain thereby permitting the use of a uniform finite element grid which provides high resolution near the wall and automatically follows boundary-layer growth. The second-order accurate Crank-Nicholson scheme is applied along with a linearization method to take advantage of the parabolic nature of the boundary-layer equations and generate a non-iterative marching routine. The SDG code can be applied to any smoothly-connected airfoil shape without modification and can be coupled to any inviscid flow solver. In this analysis, a direct viscous-inviscid interaction is accomplished between the Euler and boundary-layer codes, through the application of a transpiration velocity boundary condition. Results are presented for compressible turbulent flow past NACA 0012 and RAE 2822 airfoils at various freestream Mach numbers, Reynolds numbers, and angles of attack. All results show good agreement with experiment, and the coupled code proved to be a computationally-efficient and accurate airfoil analysis tool
Finite Difference Computing with PDEs: A Modern Software Approach
finite difference methods; programming; python; verification; numerical methods; differential equation
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