In this paper,we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyper-bolic partial differential equation of the form u...In this paper,we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyper-bolic partial differential equation of the form utt=A(x,y,t)uxx+B(x,y,t)uyy+g(x,y,t,u,ux,uy,ut),0<x,y<1,t>0 subject to appropriate initial and Dirichlet boundary conditions.We use only five evaluations of the function g and do not require any fictitious points to discretize the differential equation.The proposed method is directly applicable to wave equation in polar coordinates and when applied to a linear telegraphic hyperbolic equation is shown to be unconditionally stable.Numerical results are provided to illustrate the usefulness of the proposed method.展开更多
By using nonuniform(geometric)grid network,a new high-order finite-difference compact scheme has been obtained for the numerical solution of three-space dimensions partial differential equations of elliptic type.Singl...By using nonuniform(geometric)grid network,a new high-order finite-difference compact scheme has been obtained for the numerical solution of three-space dimensions partial differential equations of elliptic type.Single cell discretization to the elliptic equation makes it easier to compute and exhibit stability of the numerical solutions.The monotone and irreducible property of the Jacobian matrix to the system of difference equations analyses the converging behavior of the numerical solution values.As an experiment,applications of the compact scheme to Schr¨odinger equations,sine-Gordon equations,elliptic Allen–Cahn equation and Poisson’s equation have been presented with root mean squared errors of exact and approximate solution values.The results corroborate the reliability and efficiency of the scheme.展开更多
基金“The University of Delhi”under research grant No.Dean(R)/R&D/2010/1311.
文摘In this paper,we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyper-bolic partial differential equation of the form utt=A(x,y,t)uxx+B(x,y,t)uyy+g(x,y,t,u,ux,uy,ut),0<x,y<1,t>0 subject to appropriate initial and Dirichlet boundary conditions.We use only five evaluations of the function g and do not require any fictitious points to discretize the differential equation.The proposed method is directly applicable to wave equation in polar coordinates and when applied to a linear telegraphic hyperbolic equation is shown to be unconditionally stable.Numerical results are provided to illustrate the usefulness of the proposed method.
基金Science and Engineering Research Board(DST,Govt.of India)Grant No.SR/FTP/MS-020/2011Chinese Academy of Sciences President’s International Fellowship Initiative,Grant No.2015PM034。
文摘By using nonuniform(geometric)grid network,a new high-order finite-difference compact scheme has been obtained for the numerical solution of three-space dimensions partial differential equations of elliptic type.Single cell discretization to the elliptic equation makes it easier to compute and exhibit stability of the numerical solutions.The monotone and irreducible property of the Jacobian matrix to the system of difference equations analyses the converging behavior of the numerical solution values.As an experiment,applications of the compact scheme to Schr¨odinger equations,sine-Gordon equations,elliptic Allen–Cahn equation and Poisson’s equation have been presented with root mean squared errors of exact and approximate solution values.The results corroborate the reliability and efficiency of the scheme.
