Sixth-order compact finite difference method for singularly perturbed 1D reaction diffusion problems

AbstractIn this paper, the sixth-order compact finite difference method is presented for solving singularly perturbed 1D reaction–diffusion problems. The derivative of the given differential equation is replaced by finite difference approximations. Then, the given difference equation is transformed to linear systems of algebraic equations in the form of a three-term recurrence relation, which can easily be solved using a discrete invariant imbedding algorithm. To validate the applicability of the proposed method, some model examples have been solved for different values of the perturbation parameter and mesh size. Both the theoretical error bounds and the numerical rate of convergence have been established for the method. The numerical results presented in the tables and graphs show that the present method approximates the exact solution very well

Chebyshev Solution of the Nearly-Singular One-Dimensional Helmholtz Equation and Related Singular Perturbation Equations: Multiple Scale Series and the Boundary Layer Rule-of-Thumb

The one-dimensional Helmholtz equation, ε 2 u xx − u = f ( x ), arises in many applications, often as a component of three-dimensional fluids codes. Unfortunately, it is difficult to solve for ε≪1 because the homogeneous solutions are exp (± x /ε), which have boundary layers of thickness O(1/ε). By analyzing the asymptotic Chebyshev coefficients of exponentials, we rederive the Orszag–Israeli rule [16] that Chebyshev polynomials are needed to obtain an accuracy of 1% or better for the homogeneous solutions. (Interestingly, this is identical with the boundary layer rule-of-thumb in [5], which was derived for singular functions like tanh([ x −1]/ε).) Two strategies for small ε are described. The first is the method of multiple scales, which is very general, and applies to variable coefficient differential equations, too. The second, when f ( x ) is a polynomial, is to compute an exact particular integral of the Helmholtz equation as a polynomial of the same degree in the form of a Chebyshev series by solving triangular pentadiagonal systems. This can be combined with the analytic homogeneous solutions to synthesize the general solution. However, the multiple scales method is more efficient than the Chebyshev algorithm when ε is very, very tiny.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45436/1/11075_2004_Article_2865.pd

Invariant imbedding approach for the solution of the minimal surface equation

AbstractA new combined technique based on the application of a linearization procedure either (i), the combination of Outer- and Picard-approximation or (ii) the combination of Newton- and Picard-approximation, and invariant imbedding is proposed for obtaining a numerical solution of the minimal surface equation. The existence of inverses of certain matrices appearing in the invariant imbedding equations and the stability of the algorithm are investigated. The minimal surface equation under various boundary conditions and the subsonic fluid flow problem are chosen as test examples for illustrating the method. The numerical results indicate that the proposed method can be used efficiently for solving elliptic problems of a highly nonlinear nature