A High-Order Weakly L-Stable Time Integration Scheme with an Application to Burgers’ Equation

Abstract

In this paper, we propose a 7th order weakly L-stable time integration scheme. In the process of derivation of the scheme, we use explicit backward Taylor’s polynomial approximation of sixth-order and Hermite interpolation polynomial approximation of fifth order. We apply this formula in the vector form in order to solve Burger’s equation, which is a simplified form of Navier-Stokes equation. The literature survey reveals that several methods fail to capture the solutions in the presence of inconsistency and for small values of viscosity, e.g., 10−3, whereas the present scheme produces highly accurate results. To check the effectiveness of the scheme, we examine it over six test problems and generate several tables and figures. All of the calculations are executed with the help of Mathematica 11.3. The stability and convergence of the scheme are also discussed.

Weighted Average Based Differential Quadrature Method for One-Dimensional Homogeneous First Order Nonlinear Parabolic Partial Differential Equation

“…However the comparison among Table 2-Table 2 and the graphs of the numerical versus exact solution of one-dimensional Burger equation shows that the present method generates a more accurate result and it is superior to the method developed in [1], [6], [31] and It is approximate the exact solution very well.…”

“…( 1) on [0, 1] considered by Amit Kumar Verma et.al. in [6] The unique exact solution of the above 1D non linear parabolic equation (Burger equation) is given by:…”

“…In this paper, weighted average based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation (the Burger equation). To demonstrate the competence of the method, two model examples are solved by taking different values for step size h, and time step k. Numerical results obtained by the present method have been associated with numerical results obtained by the method in [1], [6], [31] and the results are summarized in Tables and graph. Moreover, in the present numerical computation, the result presented in Table 1 shows that Both root mean square error norm (L 2 ) and maximum absolute error norm (L ∞ ) are incases as the number of mesh point M and N increases in both directions.…”

“…The RMS error and maximum absolute error are calculated as in the ref. [6] given by: , | Where and are respectively exact and numerical solutions of the non linear parabolic equation at the grid point…”

“…Even though the method is capable of approximating Burger"s equation, they failed to solve a relatively small perturbation parameter. Amit [6] Present solution of burger equation using seventh order convergent weakly L-stable Newton Cotes formula and application of Burger"s equation. They used Hermit's interpolation polynomial approximation (oscillatory interpolation) and explicit backward Taylor"s polynomial approximation.…”