Richardson Extrapolation (richardson + extrapolation)

Distribution by Scientific Domains


Selected Abstracts


Finite element and finite volume simulation and error assessment of polymer melt flow in closed channels

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 11 2006
M. Vaz Jr.
Abstract This work aims at evaluating the discretization errors associated to the finite volume and finite element methods of polymer melt flow in closed channels. Two strategies are discussed: (i) Richardson extrapolation and (ii) a posteriori error estimation based on projection/smoothing techniques. The numerical model accounts for the full interaction between the thermal effects caused by viscous heating and the momentum diffusion effects dictated by a shear rate and temperature-dependent constitutive model. The simulations have been performed for the commercial polymer Polyacetal POM-M90-44. Copyright © 2006 John Wiley & Sons, Ltd. [source]


Asymptotic upper bounds for the errors of Richardson extrapolation with practical application in approximate computations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2009
Aram Soroushian
Abstract The results produced by Richardson extrapolation, though, in general, very accurate, are inexact. Numerical evaluation of this inexactness and implementation of the evaluation in practice are the objectives of this paper. First, considering linear changes of errors in the convergence plots, asymptotic upper bounds are proposed for the errors. Then, the achievement is extended to the results produced by Richardson extrapolation, and finally, an error-controlling procedure is proposed and successfully implemented in approximate computations originated in science and engineering. Copyright © 2009 John Wiley & Sons, Ltd. [source]


A numerical approach for groundwater flow in unsaturated porous media

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9-10 2006
F. Quintana
Abstract In this article, a computational tool to simulate groundwater flow in variably saturated non-deformable fractured porous media is presented, which includes a conceptual model to obtain analytical expressions of water retention and hydraulic conductivity curves for fractured hard rocks and a numerical algorithm to solve the Richards equation. To calculate effective saturation and relative hydraulic conductivity curves we adopt the Brooks,Corey model assuming fractal laws for both aperture and number of fractures. A standard Galerkin formulation was employed to solve the Richards' equation together with a Crank,Nicholson scheme with Richardson extrapolation for the time discretization. The main contribution of this paper is to group an analytical model of the authors with a robust numerical algorithm designed to solve adequately the highly non-linear Richards' equation generating a tool for porous media engineering. Copyright © 2006 John Wiley & Sons, Ltd. [source]


A least square extrapolation method for the a posteriori error estimate of the incompressible Navier Stokes problem

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2005
M. Garbey
Abstract A posteriori error estimators are fundamental tools for providing confidence in the numerical computation of PDEs. To date, the main theories of a posteriori estimators have been developed largely in the finite element framework, for either linear elliptic operators or non-linear PDEs in the absence of disparate length scales. On the other hand, there is a strong interest in using grid refinement combined with Richardson extrapolation to produce CFD solutions with improved accuracy and, therefore, a posteriori error estimates. But in practice, the effective order of a numerical method often depends on space location and is not uniform, rendering the Richardson extrapolation method unreliable. We have recently introduced (Garbey, 13th International Conference on Domain Decomposition, Barcelona, 2002; 379,386; Garbey and Shyy, J. Comput. Phys. 2003; 186:1,23) a new method which estimates the order of convergence of a computation as the solution of a least square minimization problem on the residual. This method, called least square extrapolation, introduces a framework facilitating multi-level extrapolation, improves accuracy and provides a posteriori error estimate. This method can accommodate different grid arrangements. The goal of this paper is to investigate the power and limits of this method via incompressible Navier Stokes flow computations. Copyright © 2005 John Wiley & Sons, Ltd. [source]


Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2010
Hong-Lin Liao
Abstract Alternating direction implicit (ADI) schemes are computationally efficient and widely utilized for numerical approximation of the multidimensional parabolic equations. By using the discrete energy method, it is shown that the ADI solution is unconditionally convergent with the convergence order of two in the maximum norm. Considering an asymptotic expansion of the difference solution, we obtain a fourth-order, in both time and space, approximation by one Richardson extrapolation. Extension of our technique to the higher-order compact ADI schemes also yields the maximum norm error estimate of the discrete solution. And by one extrapolation, we obtain a sixth order accurate approximation when the time step is proportional to the squares of the spatial size. An numerical example is presented to support our theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source]


Application of Richardson extrapolation to the numerical solution of partial differential equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2009
Clarence Burg
Abstract Richardson extrapolation is a methodology for improving the order of accuracy of numerical solutions that involve the use of a discretization size h. By combining the results from numerical solutions using a sequence of related discretization sizes, the leading order error terms can be methodically removed, resulting in higher order accurate results. Richardson extrapolation is commonly used within the numerical approximation of partial differential equations to improve certain predictive quantities such as the drag or lift of an airfoil, once these quantities are calculated on a sequence of meshes, but it is not widely used to determine the numerical solution of partial differential equations. Within this article, Richardson extrapolation is applied directly to the solution algorithm used within existing numerical solvers of partial differential equations to increase the order of accuracy of the numerical result without referring to the details of the methodology or its implementation within the numerical code. Only the order of accuracy of the existing solver and certain interpolations required to pass information between the mesh levels are needed to improve the order of accuracy and the overall solution accuracy. Using the proposed methodology, Richardson extrapolation is used to increase the order of accuracy of numerical solutions of the linear heat and wave equations and of the nonlinear St. Venant equations in one-dimension. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source]


An efficient high-order algorithm for solving systems of reaction-diffusion equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2002
Wenyuan Liao
Abstract An efficient higher-order finite difference algorithm is presented in this article for solving systems of two-dimensional reaction-diffusion equations with nonlinear reaction terms. The method is fourth-order accurate in both the temporal and spatial dimensions. It requires only a regular five-point difference stencil similar to that used in the standard second-order algorithm, such as the Crank-Nicolson algorithm. The Padé approximation and Richardson extrapolation are used to achieve high-order accuracy in the spatial and temporal dimensions, respectively. Numerical examples are presented to demonstrate the efficiency and accuracy of the new algorithm. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 340,354, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10012 [source]


Pricing American exchange options in a jump-diffusion model

THE JOURNAL OF FUTURES MARKETS, Issue 3 2007
Snorre Lindset
A way to estimate the value of an American exchange option when the underlying assets follow jump-diffusion processes is presented. The estimate is based on combining a European exchange option and a Bermudan exchange option with two exercise dates by using Richardson extrapolation as proposed by R. Geske and H. Johnson (1984). Closed-form solutions for the values of European and Bermudan exchange options are derived. Several numerical examples are presented, illustrating that the early exercise feature may have a significant economic value. The results presented should have potential for pricing over-the-counter options and in particular for pricing real options. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:257,273, 2007 [source]