Fourth-order numerical scheme based on half-step non-polynomial spline approximations for 1D quasi-linear parabolic equations
R.K. Mohanty1, S. Sharma2
1Department of Applied Mathematics, Faculty of Mathematics and Computer Science, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi, India
2Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, India
Keywords: квазилинейные параболические уравнения, сплайн в напряжении, обобщенное уравнение Бюргерса-Хаксли, обобщенное уравнение Бюргерса-Фишера, итерационный метод Ньютона, quasi-linear parabolic equations, spline in tension, generalized Burgers-Huxley equation, generalized Burgers-Fisher equation, Newton's iterative method
Abstract
In this article, we discuss a fourth-order accurate scheme based on non-polynomial spline in tension approximations for the solution of quasi-linear parabolic partial differential equations. The proposed numerical method requires only two half-step points and a central point on a uniform mesh in the spatial direction. This method is derived directly from a continuity condition of the first-order derivative of a non-polynomial tension spline function. The stability of the scheme is discussed using a model linear PDE. The method is directly applicable to solving singular parabolic problems in polar systems. The proposed method is tested on the generalized Burgers-Huxley equation, the generalized Burgers-Fisher equation, and Burgers' equations in polar coordinates.
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