A New Two-Level Implicit Scheme for the System of 1D Quasi-Linear Parabolic Partial Differential Equations Using Spline in Compression Approximations

• Published in: Differential equations and dynamical systems Vol. 27; no. 1-3; pp. 327 - 356
• Main Authors: Mohanty, R. K., Sharma, Sachin, Singh, Swarn
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.01.2019; Springer Nature B.V
• Subjects: Boundary conditions; Computer Science; Engineering; Finite element method; Fluid flow; High Reynolds number; Mathematical models; Mathematics; Mathematics and Statistics; Original Research; Partial differential equations; Polynomials; Reynolds number; Spline functions; Stability analysis; Generalized Burgers–Huxley equations; 65M06; Coupled Burgers’ equation; Newton’s iterative method; 65Y20; 65M22; Quasi-linear parabolic equations; Spline in compression; 65M12
• ISSN: 0971-3514; 0974-6870
• DOI: 10.1007/s12591-018-0427-5

In this article, we proposed a new two-level implicit method of accuracy two in time and four in space based on spline in compression approximations using two half-step points and a central point on a uniform mesh for the numerical solution of the system of 1D quasi-linear parabolic partial differential equations subject to appropriate initial and natural boundary conditions prescribed. The proposed method is derived directly from the continuity condition of the first order derivative of the non-polynomial compression spline function. The stability analysis for a model problem is discussed. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, we have solved generalized Burgers–Huxley equation, generalized Burgers–Fisher equation, coupled Burgers-equations and parabolic equations with singular coefficients. We show that the proposed method enables us to obtain high accurate solution for high Reynolds number.