A numerical approach to interface curves for some nonlinear diffusion equations

Abstract

Nonlinear diffusion equations in one dimensional space vt=(vm)xx+vF(v) (m> 1) appear in the fields of fluid dynamics, combustion theory, plasma physics and population dynamics. The most interesting phenomenon is the finite speed of propagation. Specifically, if the initial function has compact support, the solution has also compact support for later times, and there appears an interface between v >0 and v=O. The aim of this paper is to propose a finite difference scheme possessing the property that numerical solutions as well as numerical interfaces converge to the exact ones. Numerical examples are also presented.

Untitled

“…This is the reason why we use the irregular mesh points (3.1). Figure 1 shows the numerical interfaces (h(t) by the interface tracking scheme by Mimura, Nakaki, and Tomoeda [15], and the numerical waiting times 7)* by the present scheme (3.2)-(3.5) for 0 = 0, 1. The convergence of these numerical interfaces is proved in [15].…”

“…Figure 1 shows the numerical interfaces (h(t) by the interface tracking scheme by Mimura, Nakaki, and Tomoeda [15], and the numerical waiting times 7)* by the present scheme (3.2)-(3.5) for 0 = 0, 1. The convergence of these numerical interfaces is proved in [15]. Comparing the numerical interfaces and numerical waiting times, we can say that our numerical waiting times are reliable.…”

“…However, they do not explicitly determine the waiting time for 9 £ (4,1], In numerical points of view, there are some numerical methods to estimate the waiting time t*. Tomoeda and Mimura [18], Mimura, Nakaki, and Tomoeda [15], Gurtin, MacCamy, and Socolovsky [9], and Bertsch and Dal Passo [6] introduce interface tracking algorithms to (1.1)-(1.2), and show the numerical simulations in the case of (1.6) with 9 = 0, 9 = i, and 9=1.…”

Difference Schemes for Degenerate Parabolic Equations

“…There are a number of papers discussing difference schemes for problems like (P) when g = 0 and a = 0, e.g., [4], [11], [13], [18], [21]. In Section 5 we consider the schemes proposed in [4], which also arise from a semigroup approach to (P).…”

“…Our method also has finite speed of propagation, and in Section 6 we discuss numerical experiments conducted with both methods for a known exact solution of compact support [2], [23]. In this case, our chosen scheme was more accurate, especially in determining the interface, but not as accurate as the one-dimensional schemes in [18], [21] which are specifically designed to track the interface.…”