M. K. Jain

On the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients

Abstract Implicit difference schemes of O ( k 4 + k 2 h 2 + h 4 ), where k 0, h 0 are grid sizes in time and space coordinates respectively, are developed for the efficient numerical integration of the system of one space second order nonlinear hyperbolic equations with variable coefficients subject to appropriate initial and Dirichlet boundary conditions. The proposed difference method for a scalar equation is applied for the wave equation in cylindrical and spherical symmetry. The numerical examples are given to illustrate the fourth order convergence of the methods.

High order difference schemes for the system of two space second order nonlinear hyperbolic equations with variable coefficients

Abstract In this paper, we develop implicit difference schemes of O(k4 + k2h2 + h4), where k > 0, h > 0 are grid sizes in time and space coordinates, respectively, for solving the system of two space dimensional second order nonlinear hyperbolic partial differential equations with variable coefficients having mixed derivatives subject to appropriate initial and boundary conditions. The proposed difference method for the scalar equation is applied for the solution of wave equation in polar coordinates to obtain three level conditionally stable ADI method of O(k4 + k2h2 + h4). Some physical nonlinear problems are provided to demonstrate the accuracy of the implementation.

High accuracy difference schemes for a class of singular three space dimensional hyperbolic equations

For the numerical integration of the system of 3-D nonlinear hyperbolic equations with variable coefficients, we report two three-level implicit difference methods of 0(k 4 + k 2 h 2 + h 4) where k and h are grid sizes in time and space directions, respectively. When the coefficients of uxy, uyz and uyzare equal to zero we require only (7+19 + 7) grid points and when the coefficients of uxy, uyz and uzx are not equal to zero and the coefficients of uxx, uyy and uzz are equal we require (19+27+19) grid points. The three-level conditionally stable ADI method of 0 (k4 + k 2 h 2+ h 4) for the numerical solution of wave equation in polar coordinates is discussed. Numerical examples are provided to illustrate the methods and their fourth order convergence.

O(n3) noniterative heuristic algorithm for linear programs with error-free implementation

We present an O(n^3) noniterative heuristic algorithm and describe an implementation of p-adic arithmetic to solve the linear programming problems error-free. An optimality test as well as numerical examples depict the scope of such an algorithm. In the few cases where it did not lead to the optimal solution, it did provide a nonnegative solution of the system Ax=b, which could be used as the basic feasible solution for any one of the widely used simplex methods, say, the revised simplex algorithm without the need to include artificial variables in Ax=b for consistency check. Thus the algorithm could be at least a pre-processor for any simplex algorithm and avoids the enhancement of dimensions of the constraint matrix A and consequent additional space, computational complexities, and errors.

Fourth-order approximations at first time level, linear stability analysis and the numerical solution of multidimensional second-order nonlinear hyperbolic equations in polar coordinates

In this article, three-level implicit difference schemes of O(k4 + k2h2 + h4) where k > 0, h > 0 are grid sizes in time and space coordinates, respectively, are proposed for the numerical solution of one, two and three space-dimensional nonlinear wave equations in polar coordinates subject to appropriate initial and Dirichlet boundary conditions. We also discuss fourth-order approximation at first time level for more general case. We also obtain the stability range of the difference scheme when applied to a test equation: utt = urr + aruu − ar2u + g(r,t), a = 1 and 2 Numerical examples are provided to demonstrate the required order of convergence of the methods.

Finite difference methods of order two and four for 2-d non-linear biharmonic problems of first kind

For the numerical integration of the 2-D nonlinear biharmonic problems of first kind, we report two difference methods of order two and four over a rectangular domain. These methods use only the nine grid points and do not require fictitious points in order to approximate the boundary conditions. Derivatives of the solution are obtained as a by-product of the methods. In numerical experiments, the new second and fourth order formulas are compared with the exact solutions.

Fourth order operator splitting method for the three space parabolic equation with variable coefficients

This paper deals with two new compact difference methods of 0(k 2 + kh 2 + h 4) for solving the system of general three space non-linear parabolic equations subject to the appropriate initial and Dirichlet boundary conditions using 27 and 19 spatial grid points in the presence and absence of mixed derivative terms, respectively. We also discuss the fourth order operator splitting method for the linear equation when applied to the proposed difference method. Various numerical examples including 3-space unsteady Navier-Stokes equations are provided which demonstrate the effectiveness and the computational efficiency of the methods developed.

Preconditioned iterative methods for the nine-point approximation to the convection---diffusion equation

Iterative methods preconditioned by incomplete factorizations and sparse approximate inverses are considered for solving linear systems arising from fourth-order finite difference schemes for convection-diffusion problems. Simple recurrences for implementing the ILU(0) factorization of the nine-point scheme are derived. Different sparsity patterns are considered for computing approximate inverses for the coefficient matrix and the quality of the preconditioner is studied in terms of plots of the field of values of the preconditioned matrices. In terms of algebraic properties of the preconditioned matrices, our experimental results show that incomplete factorizations give a preconditioner of better quality than approximate inverses. Comparison of the convergence rates of GMRES applied to the preconditioned linear systems is done with respect to the field of values, Ritz and harmonic Ritz values of the preconditioned matrices. Numerical results show that the GMRES residual norm decreases rapidly when the difference between the Ritz and harmonic Ritz values becomes small. We also describe the results of experiments when some well-known Krylov subspace methods are used to solve the linear system arising from the compact fourth-order discretizations.

Solving linear differential equations as a minimum norm least squares problem with error-bounds

Linear ordinary/partial differential equations (DEs) with linear boundary conditions (BCs) are posed as an error minimization problem. This problem has a linear objective function and a system of linear algebraic (constraint) equations and inequalities derived using both the forward and the backward Taylor series expansion. The DEs along with the BCs are approximated as linear equations/inequalities in terms of the dependent variables and their derivatives so that the total error due to discretization and truncation is minimized. The total error along with the rounding errors render the equations and inequalities inconsistent to an extent or, equivalently, near-consistent, in general. The degree of consistency will be reasonably high provided the errors are not dominant. When this happens and when the equations/inequalities are compatible with the DEs, the minimum value of the total discretization and truncation errors is taken as zero. This is because of the fact that these errors could be negative as well as positive with equal probability due to the use of both the backward and forward series. The inequalities are written as equations since the minimum value of the error (implying error-bound and written/expressed in terms of a nonnegative quantity) in each equation will be zero. The minimum norm least-squares solution (that always exists) of the resulting over-determined system will provide the required solution whenever the system has a reasonably high degree of consistency. A lower error-bound and an upper error-bound of the solution are also included to logically justify the quality/validity of the solution.

An optimum generalized trapezoidal formula for the numerical integration of y′=f(x,y)

We report the determination of the parameter α in the generalized trapezoidal formula (GTF(α)) which minimizes the local truncation error.