Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHigh order numerical schemes for transport equations on bounded domains
89
0
0.0
(
0
)
Added by
Jean-Francois Coulombel
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Authors
Benjamin Boutin
Ask ChatGPT about the research
No Arabic abstract
This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes.
rate research
Read More
In this paper, we propose third-order semi-discretized schemes in space based on the tempered weighted and shifted Grunwald difference (tempered-WSGD) operators for the tempered fractional diffusion equation. We also show stability and convergence analysis for the fully discrete scheme based a Crank--Nicolson scheme in time. A third-order scheme for the tempered Black--Scholes equation is also proposed and tested numerically. Some numerical experiments are carried out to confirm accuracy and effectiveness of these proposed methods.
We develop a stable finite difference method for the elastic wave equations in bounded media, where the material properties can be discontinuous at curved interfaces. The governing equations are discretized in second order form by a fourth or sixth order accurate summation-by-parts operator. The mesh size is determined by the velocity structure of the material, resulting in nonconforming grid interfaces with hanging nodes. We use order-preserving interpolation and the ghost point technique to couple adjacent mesh blocks in an energy-conserving manner, which is supported by a fully discrete stability analysis. In numerical experiments, we demonstrate that the convergence rate is optimal, and is the same as when a globally uniform mesh is used in a single domain. In addition, with a predictor-corrector time integration method, we obtain time stepping stability with stepsize almost the same as given by the usual Courant Friedrichs Lewy condition.
We study a family of structure-preserving deterministic numerical schemes for Lindblad equations, and carry out detailed error analysis and absolute stability analysis. Both error and absolute stability analysis are validated by numerical examples.
We propose a high order numerical homogenization method for dissipative ordinary differential equations (ODEs) containing two time scales. Essentially, only first order homogenized model globally in time can be derived. To achieve a high order method, we have to adopt a numerical approach in the framework of the heterogeneous multiscale method (HMM). By a successively refined microscopic solver, the accuracy improvement up to arbitrary order is attained providing input data smooth enough. Based on the formulation of the high order microscopic solver we derived, an iterative formula to calculate the microscopic solver is then proposed. Using the iterative formula, we develop an implementation to the method in an efficient way for practical applications. Several numerical examples are presented to validate the new models and numerical methods.
We deal with the virtual element method (VEM) for solving the Poisson equation on a domain $Omega$ with curved boundaries. Given a polygonal approximation $Omega_h$ of the domain $Omega$, the standard order $m$ VEM [6], for $m$ increasing, leads to a suboptimal convergence rate. We adapt the approach of [16] to VEM and we prove that an optimal convergence rate can be achieved by using a suitable correction depending on high order normal derivatives of the discrete solution at the boundary edges of $Omega_h$, which, to retain computability, is evaluated after applying the projector $Pi^ abla$ onto the space of polynomials. Numerical experiments confirm the theory.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
