A new efficient fourth order collocation scheme for solving Burgers’ equation

Abstract

In present work a new fourth order modified cubic B-spline (mCB) based upon collocation technique (mCBCT4) has been developed to evaluate new numeric results of the nonlinear Burgers’ equation, appear in rigorous real-world physical phenomena like - sound & shock waves in viscous medium, waves in fluid filled viscous elastic tubes, magneto-hydrodynamic-waves in medium with finite electrical-conductivity, in modeling of turbulent fluid, and in continuous stochastic processes. At first, the Burgers’ equation is remodeled into a set of 1st order ordinary differential equations (ODE), in which fourth order accurate approximation of the unknown functions, and its spatial derivatives obtained via mCBCT4. In this way a set of first-order ODE is obtained, which we solve via SSP-RK(ℓ+1,ℓ) scheme (ℓ=3,4). The accuracy, efficiency and effectiveness of the developed technique mCBCT4 is demonstrated in terms of six different test examples of nonlinear Burgers’ equation by computing the error norms: L2 and L∞ errors. The proposed mCBCT4 scheme is also tested for nonlinear Burger’s equation with very small kinematic viscosities. The proposed mCBCT4 is shown unconditionally stable scheme. The numerical findings demonstrate that the developed mCBCT4 performs better than some recently developed good techniques and enables to produces comparably more accurate solutions than some recently developed reliable techniques.