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.

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In this article, we analyze and propose to compute the numerical solutions of a generalized Rosenau-KDV-RLW (Rosenau-Korteweg De Vries-Regularized Long Wave) equation based on the Haar wavelet (HW) collocation approach coupled with nonstandard finite difference (NSFD) scheme and quasilinearization. In the process of the numerical solution, the NSFD scheme is applied to discretize the first-order time derivative, Haar wavelets are applied on spatial derivatives and the non-linear term is taken care by quasilinearization technique. To discuss the efficiency of the method we compute L ∞ error and L 2 error. We also use discrete mass and energy conservation to check the accuracy of the proposed methodology. The computed results have been compared with the existing methods, for example, three-level average implicit finite difference technique, B-spline collocation, three-level linear conservative implicit finite difference scheme and conservative fourth-order stable finite difference scheme.

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In the present paper, we establish an efficient numerical scheme based on weakly L‐stable time integration convergent formula and nonstandard finite difference (NSFD) scheme. We solve Burgers' equation with Dirichlet boundary conditions as well as Neumann boundary conditions. We also solve the Fisher equation. We use Hermite approximation polynomial of order five and backward explicit Taylor's series approximation of order six to derive the numerical integration formula for the initial value problem (IVP) y′false(tfalse)=gfalse(t,yfalse),0.1emyfalse(t0false)=ρ0. We combine this method with the NSFD scheme and convert the problem into the system of algebraic equations. We discuss the convergence, truncation error, and stability of the developed method. To demonstrate the efficiency of the developed method, we compare the numerical results with some existing numerical results and exact solutions.