High-resolution compact numerical method for the system of 2D quasi-linear elliptic boundary value problems and the solu
- PDF / 8,174,894 Bytes
- 22 Pages / 595.276 x 790.866 pts Page_size
- 82 Downloads / 145 Views
ORIGINAL ARTICLE
High‑resolution compact numerical method for the system of 2D quasi‑linear elliptic boundary value problems and the solution of normal derivatives on an irrational domain with engineering applications Ishaani Priyadarshini1 · R. K. Mohanty2 Received: 15 July 2020 / Accepted: 14 August 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract In this paper, we present a novel approach to attain fourth-order approximate solution of 2D quasi-linear elliptic partial differential equation on an irrational domain. In this approach, we use nine grid points with dissimilar mesh in a single compact cell. We also discuss appropriate fourth-order numerical methods for the solution of the normal derivatives on a dissimilar mesh. The method has been protracted for solving system of quasi-linear elliptic equations. The convergence analysis is discussed to authenticate the proposed numerical approximation. On engineering applications, we solve various test problems, such as linear convection–diffusion equation, Burgers’equation, Poisson equation in singular form, NS equations, bi- and tri-harmonic equations and quasi-linear elliptic equations to show the efficiency and accuracy of the proposed methods. A comprehensive comparative computational experiment shows the accuracy, reliability and credibility of the proposed computational approach. Keywords Quasi-linear elliptic equations · Dissimilar mesh · Irrational domain · Fourth-order approximation · Normal derivatives · Error analysis · Burgers’ equation · Convection–diffusion equation · Bi- and tri-harmonic equations · Navier– Stokes equations of motion Mathematics subject classification 65N06 · 65N12
1 Introduction and mathematical background The two-dimensional quasi-linear elliptic partial differential equation (PDE) is given by ( ) a(x, y, u)uxx + b(x, y, u)uyy = 𝜙 x, y, u, ux , uy ,
0 < x < xa , 0 < y < yb , * R. K. Mohanty [email protected] Ishaani Priyadarshini [email protected] 1
Department of Electrical and Computer Science, University of Delaware, Newark, DE 19716, USA
Department of Applied Mathematics, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi 110021, India
2
(1)
where xa and yb are real numbers, and (xa ∕yb ) is an irrational number. We define: A rectangular domain is said to be irrational, if the ratio of two non-parallel sides is irrational. The Dirichlet type boundary conditions are given by (2)
u(x, y) = u0 (x, y), (x, y) ∈ Γ,
where{ Γ is the boundary of the } solution domain Ωh = (x, y)||0 < x < xa , 0 < y < yb . Equation (1) is assumed to satisfy the ellipticity condition a(x, y, u).b(x, y, u) > 0 in Ωh . The elliptic PDE of type (1) model many phenomenal importance problems like the singular Poisson equation, diffusion–convection equation, nonlinear Burgers’ and viscous Navier–Stokes (NS) equations of motion in polar coordinates. We assume that for the boundary value problem (BVP) (1) and (2): 1. u(x, y) ∈ C6 and a(x, y, u), b(x, y, u) ∈ C4,
13
Vol.:(0123456789)
( ) ≥ 0, 2. ϕ is con
Data Loading...
