scholarly journals Block Hybrid k-Step Backward Differentiation Formulas for Large Stiff Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
S. N. Jator ◽  
E. Agyingi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stiff Systems ◽  
Partial Differential ◽  
Differentiation Formula ◽  
Block Scheme ◽  
Backward Differentiation Formulas ◽  
Large Systems ◽  
Grid Points ◽  
Differentiation Formulas

This paper presents a generalized high order block hybrid k-step backward differentiation formula (HBDF) for solving stiff systems, including large systems resulting from the semidiscretization parabolic partial differential equations (PDEs). A block scheme in which two off-grid points are specified by the zeros of the second degree Chebyshev polynomial of the first kind is examined for convergence, L and A stabilities. Numerical simulations that illustrate the accuracy of a Chebyshev based method are given for selected stiff systems and partial differential equations.

scholarly journals On the Multidomain Bivariate Spectral Local Linearisation Method for Solving Systems of Nonsimilar Boundary Layer Partial Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Vusi Mpendulo Magagula
Keyword(s):  
Boundary Layer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Large Time ◽  
Nonlinear Partial Differential Equations ◽  
Computation Time ◽  
General System ◽  
Partial Differential ◽  
Boundary Layer Equations ◽  
Grid Points

In this work, a novel approach for solving systems of nonsimilar boundary layer equations over a large time domain is presented. The method is a multidomain bivariate spectral local linearisation method (MD-BSLLM), Legendre-Gauss-Lobatto grid points, a local linearisation technique, and the spectral collocation method to approximate functions defined as bivariate Lagrange interpolation. The method is developed for a general system of n nonlinear partial differential equations. The use of the MD-BSLLM is demonstrated by solving a system of nonlinear partial differential equations that describe a class of nonsimilar boundary layer equations. Numerical experiments are conducted to show applicability and accuracy of the method. Grid independence tests establish the accuracy, convergence, and validity of the method. The solution for the limiting case is used to validate the accuracy of the MD-BSLLM. The proposed numerical method performs better than some existing numerical methods for solving a class of nonsimilar boundary layer equations over large time domains since it converges faster and uses few grid points to achieve accurate results. The proposed method uses minimal computation time and its accuracy does not deteriorate with an increase in time.

scholarly journals On the Integration of Stiff ODEs Using Block Backward Differentiation Formulas of Order Six

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 952
Author(s):  
Amiratul Ashikin Nasarudin ◽  
Zarina Bibi Ibrahim ◽  
Haliza Rosali
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Similar Type ◽  
Differentiation Formula ◽  
First Order ◽  
Backward Differentiation Formula ◽  
Fully Implicit ◽  
Stiff Odes ◽  
Backward Differentiation Formulas ◽  
Differentiation Formulas

In this research, a six-order, fully implicit Block Backward Differentiation Formula with two off-step points (BBDFO(6)), for the integration of first-order ordinary differential equations (ODEs) that exhibit stiffness, is proposed. The order, consistency and stability properties of the method are discussed, and the method is found to be zero stable and consistent. Hence, the method is convergent. The numerical comparisons with the existing methods of a similar type are given to demonstrate the accuracy of the derived method.

APPROXIMATE SOLUTION OF MODIFIED CAMASSA–HOLM AND DEGASPERIS–PROCESI EQUATIONS USING WAVELET OPTIMIZED FINITE DIFFERENCE METHOD

2013 ◽  
Vol 11 (02) ◽  
pp. 1350019
Author(s):  
RATIKANTA BEHERA ◽  
MANI MEHRA
Keyword(s):  
Approximate Solution ◽  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Grid Points ◽  
Scientific Phenomena

In this paper, we apply wavelet optimized finite difference method to solve modified Camassa–Holm and modified Degasperis–Procesi equations. The method is based on Daubechies wavelet with finite difference method on an arbitrary grid. The wavelet is used at regular intervals to adaptively select the grid points according to the local behaviour of the solution. The purpose of wavelet-based numerical methods for solving linear or nonlinear partial differential equations is to develop adaptive schemes, in order to achieve accuracy and computational efficiency. Since most of physical and scientific phenomena are modeled by nonlinear partial differential equations, but it is difficult to handle nonlinear partial differential equations analytically. So we need approximate solution to solve these type of partial differential equation. Numerical results are presented for approximating modified Camassa–Holm and modified Degasperis–Procesi equations, which demonstrate the advantages of this method.

A New Fourth-Order Compact Off-Step Discretization for the System of 2D Nonlinear Elliptic Partial Differential Equations

2012 ◽  
Vol 2 (1) ◽  
pp. 59-82 ◽  
Author(s):  
R. K. Mohanty ◽  
Nikita Setia
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourth Order ◽  
Elliptic Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Polar Coordinates ◽  
Partial Differential ◽  
Nonlinear Elliptic ◽  
Physical Problems ◽  
Grid Points

AbstractThis paper discusses a new fourth-order compact off-step discretization for the solution of a system of two-dimensional nonlinear elliptic partial differential equations subject to Dirichlet boundary conditions. New methods to obtain the fourth-order accurate numerical solution of the first order normal derivatives of the solution are also derived. In all cases, we use only nine grid points to compute the solution. The proposed methods are directly applicable to singular problems and problems in polar coordinates, which is a main attraction. The convergence analysis of the derived method is discussed in detail. Several physical problems are solved to demonstrate the usefulness of the proposed methods.

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