The author proves the existence of the asymptotic error expansion to the Peaceman-Rachford finite-difference scheme for the first boundary value problem of the two-dimensional evolationary equation on the so-called uniform and nearly uniform domains. This expansion leads, by Richardson extrapolation, to a simple process for accelerating the convergence of the method. A numerical example is given.
We consider a family of conforming finite element schemes with piecewise polynomial space of degree in space for solving the wave equation, as a model for second order hyperbolic equations. The discretization in time is performed using the Newmark method. A new a priori estimate is proved. Thanks to this new a priori estimate, it is proved that the convergence order of the error is in the discrete norms of and , where and are the mesh size of the spatial and temporal discretization, respectively....
In order to accommodate solutions with multiple phases, corresponding to crossing rays, we formulate geometrical optics for the scalar wave equation as a kinetic transport equation set in phase space. If the maximum number of phases is finite and known a priori we can recover the exact multiphase solution from an associated system of moment equations, closed by an assumption on the form of the density function in the kinetic equation. We consider two different closure assumptions based on delta...
There are many methods and approaches to solving convection--diffusion problems. For those who want to solve such problems the situation is very confusing and it is very difficult to choose the right method. The aim of this short overview is to provide basic guidelines and to mention the common features of different methods. We place particular emphasis on the concept of linear and non-linear stabilization and its implementation within different approaches.
In this article we discuss some issues related to Air Pollution modelling (as viewed by the authors): subgrid parametrization, multiphase modelling, reduction of high dimensional models and data assimilation. Numerical applications are given with POLAIR, a 3D numerical platform devoted to modelling of atmospheric trace species.
Numerical experiments in J. Maubach: Local bisection refinement and optimal order algebraic multilevel preconditioners, PRISM-97 conference Proceedings, 1977, 121–136 indicated that the refinement with the use of local bisections presented in J. Maubach: Local bisection refinement for -simplicial grids generated by reflections, SIAM J. Sci. Comput. 16 (1995), 210–227 leads to highly locally refined computational 2-meshes which can be very efficiently load-balanced with the use of a space-filling...
The paper presents the theory of the discontinuous Galerkin finite element method for the space-time discretization of a linear nonstationary convection-diffusion-reaction initial-boundary value problem. The discontinuous Galerkin method is applied separately in space and time using, in general, different nonconforming space grids on different time levels and different polynomial degrees and in space and time discretization, respectively. In the space discretization the nonsymmetric interior...
The paper is concerned with the application of the space-time discontinuous Galerkin method (STDGM) to the numerical solution of the interaction of a compressible flow and an elastic structure. The flow is described by the system of compressible Navier-Stokes equations written in the conservative form. They are coupled with the dynamic elasticity system of equations describing the deformation of the elastic body, induced by the aerodynamical force on the interface between the gas and the elastic...
This course gives an introduction to the design of efficient datatypes for adaptive wavelet-based applications. It presents some code fragments and benchmark technics useful to learn about the design of sparse data structures and adaptive algorithms. Material and practical examples are given, and they provide good introduction for anyone involved in the development of adaptive applications. An answer will be given to the question: how to implement and efficiently use the discrete wavelet transform...
This paper is concerned with the analysis and implementation of spectral Galerkin methods for a class of Fokker-Planck equations that arises from the kinetic theory of dilute polymers. A relevant feature of the class of equations under consideration from the viewpoint of mathematical analysis and numerical approximation is the presence of an unbounded drift coefficient, involving a smooth convex potential that is equal to along the boundary of the computational domain . Using a symmetrization...
This paper is concerned with the analysis and implementation of spectral Galerkin methods for a class of Fokker-Planck equations that arises from the kinetic theory of dilute polymers. A relevant feature of the class of equations under consideration from the viewpoint of mathematical analysis and numerical approximation is the presence of an unbounded drift coefficient, involving a smooth convex potential U that is equal to +∞ along the boundary ∂D of the computational domain D. Using a symmetrization...
This work presents simulations of incompressible fluid flow interacting with a moving rigid body. A numerical algorithm for incompressible Navier-Stokes equations in a general coordinate system is applied to two types of body motion, prescribed and flow-induced. Discretization in spatial coordinates is based on the spectral/hp element method. Specific techniques of stabilisation, mesh design and approximation quality estimates are described and compared. Presented data show performance of the solver...
We consider here the Interior Penalty Discontinuous Galerkin (IPDG) discretization of the wave equation. We show how to derive the optimal penalization parameter involved in this method in the case of regular meshes. Moreover, we provide necessary stability conditions of the global scheme when IPDG is coupled with the classical Leap–Frog scheme for the time discretization. Numerical experiments illustrate the fact that these conditions are also sufficient.