In this article, we study the a posteriori H1 and L2 error estimates for Crouzeix-Raviart nonconforming finite volume element discretization of general second-order elliptic problems in [Ropf]2. The error estimators yield global upper and local lower bounds. Finally, numerical experiments are performed to illustrate the theoretical findings. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009
miércoles, 16 de septiembre de 2009
A posteriori error analysis of nonconforming finite volume elements for general second-order elliptic PDEs
In this article, we study the a posteriori H1 and L2 error estimates for Crouzeix-Raviart nonconforming finite volume element discretization of general second-order elliptic problems in [Ropf]2. The error estimators yield global upper and local lower bounds. Finally, numerical experiments are performed to illustrate the theoretical findings. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009
Bernstein Ritz-Galerkin method for solving an initial-boundary value problem that combines Neumann and integral condition for the wave equation
In this article, the Ritz-Galerkin method in Bernstein polynomial basis is implemented to give an approximate solution of a hyperbolic partial differential equation with an integral condition. We will deal here with a type of nonlocal boundary value problem, that is, the solution of a hyperbolic partial differential equation with a nonlocal boundary specification. The nonlocal conditions arise mainly when the data on the boundary cannot be measured directly. The properties of Bernstein polynomial and Ritz-Galerkin method are first presented, then Ritz-Galerkin method is used to reduce the given hyperbolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique presented in this article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009
Finite volume simulation of waves formed by sliding masses
We numerically study a relatively simple two-dimensional (2D) model for landslide-generated nonlinear surface water waves. The landslides are modeled as rigid and impervious bodies translating on a flat or an inclined bottom. The water motion is assumed to be properly modeled by the 2D nonlinear system of shallow water equations. The resulting 2D system is numerically solved by means of a conservative well-balanced high-resolution finite volume upwind scheme esspecially adapted to treat advancing wet/dry fronts over irregular topography. Numerical results for 1D and 2D benchmark cases include comparisons with analytical or asymptotic solutions as well as comparisons with experimental data. The numerical investigation reveals that although the presented model has certain limitations, it appears to be able to model important aspects and the most significant characteristics of wave formation and propagation in their initial generation stage, namely, the waves moving toward the shore, the subsequent run-up and run-down, the waves propagating toward deep water, as well as the shape and arrival time of these waves. Copyright © 2009 John Wiley & Sons, Ltd.
Implicit time integration of hyperbolic conservation laws via discontinuous Galerkin methods
A high-order, matrix-free implicit method has been developed for the transient solutions of hyperbolic conservation laws. The discontinuous Galerkin method is applied for temporal discretization. This method has the advantage that its discretization error is [Oscr]([Delta]t2p+1) when a polynomial basis of degree p is used for time discretization. The nonlinear system of equations from the implicit time discretization is solved at each time step using a nonlinear Krylov subspace projection method. The system of linear equations is solved by the generalized minimum residual algorithm with a lower-upper symmetric Gauss-Seidel preconditioner. The numerical results from the inviscid Burgers' equation indicate that the implicit method is several times faster in performance relative to explicit integration by the total variation diminishing Runge-Kutta method of order 3. The forward Euler method requires a time step proportional to the square of the spatial step for stability with equations such as Burgers' equation (J. Sci. Comput. 2001; 16:173-261). It would, hence, be much less efficient than other explicit methods, for example, Cockburn and Shu (Math. Comput. 1989; 52:411-435), which would only require a time step proportional to the spatial step. Copyright © 2009 John Wiley & Sons, Ltd.
Five-point positive meshless scheme for hyperbolic conservation laws
We consider a five-point positive meshless collocation method for the numerical solutions of transport process described by hyperbolic conservation laws. This positive meshless method uses the five-point scheme approximation for derivatives, and an artificial dissipation term to ensure the positivity of coefficients. The numerical examples confirm the good performance of the present five-point positive meshless scheme. Copyright © 2009 John Wiley & Sons, Ltd.
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