Navnit Jha

A class of variable mesh spline in compression methods for singularly perturbed two point singular boundary value problems

We report numerical techniques for a class of singularly perturbed two point singular boundary value problems on a non-uniform mesh using spline in compression. The proposed methods are stable everywhere in the solution region including the vicinity of the singularity. Error analysis of a method is briefly discussed. Numerical results are provided to illustrate the proposed methods.

Spline in compression method for the numerical solution of singularly perturbed two-point singular boundary-value problems

In this article, using spline in compression, we discuss three difference schemes for the numerical solution of singularly perturbed two-point singular boundary-value problems. The proposed schemes are second-order accurate and applicable to problems both in singular and non-singular cases. Convergence analysis of a difference scheme is discussed and numerical results are given to illustrate the utility of the proposed methods.

A fifth order accurate geometric mesh finite difference method for general nonlinear two point boundary value problems

Second order boundary value problems are treated using fifth order accurate geometric mesh finite difference approximations. The method uses evaluation at two off steps, three neighbouring knots and it reduces to simple five terms recurrence relations. The formal convergence analysis and error estimate reveals that the truncation errors of the new discretizations are fifth order accurate. Some physical problems are solved to demonstrate efficiency and reliability of the proposed numerical method. The root mean square errors for different values of mesh ratio parameter have been given in tables to support the utility of the method.

A Fifth (Six) Order Accurate, Three-Point Compact Finite Difference Scheme for the Numerical Solution of Sixth Order Boundary Value Problems on Geometric Meshes

Two point boundary value problems for sixth order, mildly nonlinear ordinary differential equations, are encountered in various areas of science and technology. A three-point, compact finite difference scheme for solving such problems is presented. The sixth order differential equation is treated as system of three second order equations. The scheme described can be viewed as a generalization of the Numerov-type scheme of Chawla (IMA J Appl Math 24:35–42, 1979). It is fifth order accurate on geometric meshes (non-uniform), and sixth order accurate on uniform meshes. It is applicable both to nonsingular and singular problems. Theoretical error bounds are derived and the convergence is proven. Numerical tests confirm the theoretical predictions.

An exponential expanding meshes sequence and finite difference method adopted for two-dimensional elliptic equations

We demonstrate a new nonuniform mesh finite difference method to obtain accurate solutions for the elliptic partial differential equations in two dimensions with nonlinear first-order partial derivative terms. The method will be based on a geometric grid network area and included among the most stable differencing scheme in which the nine-point spatial finite differences are implemented, thus arriving at a compact formulation. In general, a third order of accuracy has been achieved and a fourth-order truncation error in the solution values will follow as a particular case. The efficiency of using geometric mesh ratio parameter has been shown with the help of illustrations. The convergence of the scheme has been established using the matrix analysis, and irreducibility is proved with the help of strongly connected characteristics of the iteration matrix. The difference scheme has been applied to test convection diffusion equation, steady state Burger’s equation, ocean model and a semi-linear elliptic equation. The computational results confirm the theoretical order and accuracy of the method.

New Nonpolynomial Spline in Compression Method of for the Solution of 1D Wave Equation in Polar Coordinates

We propose a three-level implicit nine point compact finite difference formulation of order two in time and four in space direction, based on nonpolynomial spline in compression approximation in -direction and finite difference approximation in -direction for the numerical solution of one-dimensional wave equation in polar coordinates. We describe the mathematical formulation procedure in detail and also discussed the stability of the method. Numerical results are provided to justify the usefulness of the proposed method.

Geometric Mesh Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System

Numerical method based on three geometric stencils has been proposed for the numerical solution of nonlinear singular fourth-order ordinary differential equations. The method can be easily extended to the sixth-order differential equations. Convergence analysis proves the third-order convergence of the proposed scheme. The resulting difference equations lead to block tridiagonal matrices and can be easily solved using block Gauss-Seidel algorithm. The computational results are provided to justify the usefulness and reliability of the proposed method.

A Second Order Non-uniform Mesh Discretization for the Numerical Treatment of Singular Two-Point Boundary Value Problems with Integral Forcing Function

In the present work, we examine the three-point numerical scheme for the non-linear second order ordinary differential equations having integral form of forcing function. The approximations of solution values are obtained by means of finite difference scheme based on a special type of non-uniform meshes. The derivatives as well as integrals are approximated with simple second order accuracy both on uniform meshes and non-uniform meshes. A brief convergence analysis based on irreducible and monotone behaviour of Jacobian matrix to the numerical scheme is provided. The scheme is then tested on linear and non-linear examples that justify the order and accuracy of the new method.

An O(h5k) accurate finite difference method for the numerical solution of fourth order two point boundary value problems on geometric meshes

An O(hk5) accurate finite difference method for the numerical solution of fourth order two point boundary value problems on geometric meshe

A third (four)-order accurate nine-point compact EEM-FDM for coupled system of mildly non-linear elliptic equations

In this paper, we formulate a new algorithm for the approximate solution values of coupled mildly non-linear elliptic boundary value problems in two-space dimensions. The method is obtained on a non-uniformly spaced mesh points in such a manner that fourth-order accuracy of the scheme is a particular case of third-order method. The special nine-point discretization makes it easier to solve and applicable to the problems possessing layer behaviour. Applications to stationary Navier-Stokes equation and biharmonic equations have been given to illustrate the efficiency and robustness of the scheme.