Analysis and Implementation of High-order Compact Finite Difference Schemes - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Analysis and Implementation of High-order Compact Finite Difference Schemes write by Jonathan Tyler. This book was released on 2007. Analysis and Implementation of High-order Compact Finite Difference Schemes available in PDF, EPUB and Kindle. The derivation of centered compact schemes at interior and boundary grid points is performed and an analysis of stability and computational efficiency is given. Compact schemes are high order implicit methods for numerical solutions of initial and/or boundary value problems modeled by differential equations. These schemes generally require smaller stencils than the traditional explicit finite difference counterparts. To avoid numerical instabilities at and near boundaries and in regions of mesh non-uniformity, a numerical filtering technique is employed. Experiments for non-stationary linear problems (convection, heat conduction) and also for nonlinear problems (Burgers' and KdV equations) were performed. The compact solvers were combined with Euler and fourth-order Runge-Kutta time differencing. In most cases, the order of convergence of the numerical solution to the exact solution was the same as the formal order of accuracy of the compact schemes employed.
Analysis and Development of Compact Finite Difference Schemes with Optimized Numerical Dispersion Relation
Analysis and Development of Compact Finite Difference Schemes with Optimized Numerical Dispersion Relation - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Analysis and Development of Compact Finite Difference Schemes with Optimized Numerical Dispersion Relation write by Yi-Hung Kuo. This book was released on 2014. Analysis and Development of Compact Finite Difference Schemes with Optimized Numerical Dispersion Relation available in PDF, EPUB and Kindle. Finite difference approximation, in addition to Taylor truncation errors, introduces numerical dispersion-and-dissipation errors into numerical solutions of partial differential equations. We analyze a class of finite difference schemes which are designed to minimize these errors (at the expanse of formal order of accuracy), and we give a quantitative analysis of the interplay between the Taylor truncation errors and the dispersion-and-dissipation errors when refining meshes. In particular, we study the numerical dispersion relation of the fully discretized non-dispersive transport equation in one and multi-dimensions. We derive the numerical phase error and the L 2 -norm error of the solution in terms of the dispersion-and-dissipation error. Based on our analysis, we investigate the error dynamics among various optimized compact schemes and the unoptimized higher-order generalized Pad\'e compact schemes, taking into account four important factors, namely, (i) error tolerance, (ii) computer memory capacity, (iii) resolvable wavenumber, and (iv) CPU/GPU time. The dynamics shed light on the principles of designing suitable optimized compact schemes for a given problem. Using these principles as guidelines, we then propose an optimized scheme that prescribes the numerical dispersion relation before finding the corresponding discretization. This approach produces smaller numerical dispersion-and-dissipation errors for linear and nonlinear problems, compared with the unoptimized higher-order compact schemes and other optimized schemes developed in the literature. Finally, we discuss the difficulty of developing an optimized composite boundary scheme for problems with non-trivial boundary conditions. We propose a composite scheme that introduces a buffer zone to connect an optimized interior scheme and an unoptimized boundary scheme. Our numerical experiments show that this strategy produces small L2-norm error when a wave packet passes through the non-periodic boundary.
Finite Difference Schemes and Partial Differential Equations
Finite Difference Schemes and Partial Differential Equations - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Finite Difference Schemes and Partial Differential Equations write by John C. Strikwerda. This book was released on 1989-09-28. Finite Difference Schemes and Partial Differential Equations available in PDF, EPUB and Kindle.
Time-stable Boundary Conditions for Finite-difference Schemes Solving Hyperbolic Systems: Methodology and Application to High-order Compact Schemes
Time-stable Boundary Conditions for Finite-difference Schemes Solving Hyperbolic Systems: Methodology and Application to High-order Compact Schemes - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Time-stable Boundary Conditions for Finite-difference Schemes Solving Hyperbolic Systems: Methodology and Application to High-order Compact Schemes write by Mark H. Carpenter. This book was released on 1993. Time-stable Boundary Conditions for Finite-difference Schemes Solving Hyperbolic Systems: Methodology and Application to High-order Compact Schemes available in PDF, EPUB and Kindle.
High-Order Compact Schemes for Parabolic Problems with Mixed Derivatives in Multiple Space Dimensions
High-Order Compact Schemes for Parabolic Problems with Mixed Derivatives in Multiple Space Dimensions - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook High-Order Compact Schemes for Parabolic Problems with Mixed Derivatives in Multiple Space Dimensions write by Bertram Düring. This book was released on 2014. High-Order Compact Schemes for Parabolic Problems with Mixed Derivatives in Multiple Space Dimensions available in PDF, EPUB and Kindle. We present a high-order compact finite difference approach for a rather general class of parabolic partial differential equations with time and space dependent coefficients as well as with mixed second-order derivative terms in n spatial dimensions. Problems of this type arise frequently in computational fluid dynamics and computational finance. We derive general conditions on the coefficients which allow us to obtain a high-order compact scheme which is fourth-order accurate in space and second-order accurate in time. Moreover, we perform a thorough von Neumann stability analysis of the Cauchy problem in two and three spatial dimensions for vanishing mixed derivative terms, and also give partial results for the general case. The results suggest unconditional stability of the scheme. As an application example we consider the pricing of European Power Put Options in the multidimensional Black-Scholes model for two and three underlying assets. Due to the low regularity of typical initial conditions we employ the smoothing operators of Kreiss et al. to ensure high-order convergence of the approximations of the smoothed problem to the true solution.
