Swarn Singh

A new spline in compression approximation for one space dimensional quasilinear parabolic equations on a variable mesh

In this paper, we propose a new two level implicit method of order two in time and four in space directions, based on spline in compression approximation for the numerical solution of one space dimensional quasi-linear parabolic partial differential equation on a uniform mesh. The derivation and the stability of the proposed method are discussed in details. We have extended the method to non-uniform mesh. Numerical results are given to illustrate the usefulness of the proposed method.

A new coupled approach high accuracy numerical method for the solution of 3D non-linear biharmonic equations

In this paper, we derive a new fourth order finite difference approximation based on arithmetic average discretization for the solution of three-dimensional non-linear biharmonic partial differential equations on a 19-point compact stencil using coupled approach. The numerical solutions of unknown variable u(x,y,z) and its Laplacian @?^2u are obtained at each internal grid point. The resulting stencil algorithm is presented which can be used to solve many physical problems. The proposed method allows us to use the Dirichlet boundary conditions directly and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. The new method is tested on three problems and the results are compared with the corresponding second order approximation, which we also discuss using coupled approach.

A new algorithm based on spline in tension approximation for 1D quasi-linear parabolic equations on a variable mesh

In this paper, we propose a new two-level implicit method of order two in time and four in space directions, based on spline in tension approximation for the numerical solution of one space dimensional quasi-linear parabolic partial differential equation on a uniform mesh. We have discussed the derivation of the proposed method in detail and have also discussed the stability analysis for a model problem. We have extended the method to non-uniform mesh. Numerical results are given to illustrate the usefulness of the proposed methods.

Numerical solution of second-order one-dimensional hyperbolic equation by exponential B-spline collocation method

In this paper, we propose a method based on collocation of exponential B-splines to obtain numerical solution of a nonlinear second-order one-dimensional hyperbolic equation subject to appropriate initial and Dirichlet boundary conditions. The method is a combination of B-spline collocation method in space and two-stage, second-order strong-stability-preserving Runge–Kutta method in time. The proposed method is shown to be unconditionally stable. The efficiency and accuracy of the method are successfully described by applying the method to a few test problems.