Journal ArticleDOI
On the exponentially slow motion of a viscous shock
Luis G. Reyna,Michael J. Ward +1 more
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In this paper, the authors studied the internal layer behavior associated with the following viscous shock problem in the limit e 0: ==================\/\/\/\/\/\/£££€££ £££•££'€£• ££ £•£•€£ £'££ • ££•$££·££Abstract:
Using formal asymptotic methods, we study the internal layer behavior associated with the following viscous shock problem in the limit e 0:
The convex nonlinearity f(u) satisfies f(α) = f(–α) For the steady problem, we show that the method of matched asymptotic expansions fails to uniquely determine the location of the equilibrium shock layer solution This indeterminacy, resulting from neglecting certain exponentially small effects, is eliminated by using the projection method, which exploits certain properties of the spectrum associated with the linearized operator For the time dependent problem, we show that the viscous shock, which is formed from initial data, drifts towards the equilibrium solution on an exponentially long time interval of the order O(eC/e), for some C > 0 This exponentially slow behavior is analyzed by deriving an equation of motion for the location of the viscous shock For Burgers equation (f(u) = u2/2), the results give an analytical characterization of the slow shock layer motion observed numerically in Kreiss and Kreiss; see [11] We also show that the shock layer behavior is very sensitive to small changes in the boundary operator In addition, using a WKB-type method, the slow viscous shock motion is studied numerically for small e, the results comparing favorably with corresponding analytical results Finally, we relate the slow viscous shock motion to similar slow internal layer motion for the Allen-Cahn equationread more
Citations
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Adaptive residual subsampling methods for radial basis function interpolation and collocation problems
Tobin A. Driscoll,Alfa Heryudono +1 more
TL;DR: This work constructs a new adaptive algorithm for radial basis functions (RBFs) method applied to interpolation, boundary-value, and initial-boundary-value problems with localized features.
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The keller-segel model with logistic sensitivity function and small diffusivity
Yasmin Dolak,Christian Schmeiser +1 more
TL;DR: Numerical and analytic evidence indicates that solutions of this problem converge to irregular patterns of cell aggregates separated by entropic shocks from vacuum regions as time tends to infinity.
Journal ArticleDOI
Supersensitivity due to uncertain boundary conditions
TL;DR: In this paper, the viscous Burgers' equation subject to perturbations on the boundary conditions is studied, and two kinds of perturbation are considered: deterministic and random.
Journal ArticleDOI
A systematic literature review of Burgers’ equation with recent advances
TL;DR: The objectives of this paper are to discuss the recent developments in mathematical modelling of Burgers’ equation, and throw some light on the plethora of challenges which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a numerical simulation of ordinary / partial differential equations.
Journal ArticleDOI
Shock layer movement for Burgers' equation
TL;DR: The result suggests a corresponding asymptotic approach which provides the shock layer location $x_\epsilon ( t )$ as the solution of an initial value problem and shows how the steady-state shock location can be changed substantially by asymPTotic exponentially small perturbations.
References
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Journal ArticleDOI
On a quasi-linear parabolic equation occurring in aerodynamics
TL;DR: In this paper, the Navier-Stokes equations for one-dimensional non-stationary flow of a compressible viscous fluid are compared to the shock wave theory of a model of turbulence.
Journal ArticleDOI
A collocation solver for mixed order systems of boundary value problems
TL;DR: Implementation of a spline collocation method for solving boundary value problems for mixed order systems of ordinary differential equations is discussed and the resulting general purpose code, COLSYS, is tested on a number of examples to demonstrate its stability, efficiency and flexibility.
Journal ArticleDOI
Metastable patterns in solutions of ut = ϵ2uxx − f(u)
Jack Carr,Robert L. Pego +1 more
TL;DR: In this article, the authors consider the problem of finding a pattern of interfacial layers that persist for exponentially long times proportional to exp{A±l/ϵ, where A = F(±1) and l is the minimum distance between layers.
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Slow-motion manifolds, dormant instability, and singular perturbations
G. Fusco,Jack K. Hale +1 more
TL;DR: In this article, the coexistence of two phases at the transition temperature is kept under observation for a long time, and it is observed that the system is not exactly in equilibrium and a very slow evolution driven by surface tension is taking place.
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