1984
DOI: 10.1007/bf03167863
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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 s… Show more

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Cited by 25 publications
(14 citation statements)
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“…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].…”
Section: Numericalmentioning
confidence: 99%
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“…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].…”
Section: Numericalmentioning
confidence: 99%
“…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.…”
Section: Numericalmentioning
confidence: 99%
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“…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).…”
mentioning
confidence: 99%
“…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.…”
mentioning
confidence: 99%