Stability and Computation of Dynamic Patterns in PDEs

Beyn, Wolf–Jürgen; Otten, Denny; Rottmann-Matthes, Jens

doi:10.1007/978-3-319-01300-8_3

Cited by 9 publications

(34 citation statements)

References 56 publications

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“…For initial data u 0 close to a wave we expect v(·, t) → v , µ(t) → µ as t → ∞. Travelling waves in parabolic systems and their stability are analysed in [35,55,62,59], and numerical applications of the freezing method for this case appear in in [10]. Next, consider the parabolic system (1.12) in several space dimensions.…”

Section: Application To Evolution Equationsmentioning

confidence: 99%

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Computation and stability of waves in equivariant evolution equations

Beyn

Otten

2019

EMS Series of Congress Reports

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Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant evolution equation. In numerical computations the freezing method takes advantage of this structure by splitting the evolution of the PDE into the dynamics on the underlying Lie group and on some reduced phase space. The approach raises a series of questions which were answered to a certain degree by the project: linear stability implies non-linear (asymptotic) stability, persistence of stability under discretisation, analysis and computation of spectral structures, first versus second order evolution systems, well-posedness of partial differential algebraic equations, spatial decay of wave profiles and truncation to bounded domains, analytical and numerical treatment of wave interactions, relation to connecting orbits in dynamical systems. A further numerical problem related to this topic will be discussed, namely the solution of non-linear eigenvalue problems via a contour method.

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Section: Application To Evolution Equationsmentioning

confidence: 99%

“…is called the quintic Nagumo equation (short: QNE), [10]. .1), parameters b j and spatial domain as before, initial data v 0 = u 0 , and the templatev = u 0 on the time range [0, 3000].…”

Section: Applications To Parabolic Hyperbolic and Hamiltonian Systemsmentioning

confidence: 99%

Computation and stability of waves in equivariant evolution equations

Beyn

Otten

2019

EMS Series of Congress Reports

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“…For more details on the computation, see [26,Section 10.3]. Some general theory and applications of the freezing method may be found in [7,8,9,10,37]. Note that in case d = 2 we have a clockwise rotation, if S 12 > 0, and a counter clockwise rotation, if S 12 < 0.…”

Section: Rotating Waves In Reaction Diffusion Systemsmentioning

confidence: 99%

“…Using these solutions as initial data, the freezing method from [7,8] provides an approximate rotating wave with profile w in the following format…”

Section: Rotating Waves In Reaction Diffusion Systemsmentioning

confidence: 99%

Spatial decay of rotating waves in reaction diffusion systems

Beyn

Otten

2016

Dynamics of Partial Differential Equations

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Abstract. In this paper we study nonlinear problems for Ornstein-Uhlenbeck operatorswhere the matrix A ∈ R N,N is diagonalizable and has eigenvalues with positive real part, the map f : R N → R N is sufficiently smooth and the matrix S ∈ R d,d in the unbounded drift term is skew-symmetric. Nonlinear problems of this form appear as stationary equations for rotating waves in time-dependent reaction diffusion systems. We prove under appropriate conditions that every bounded classical solution v of the nonlinear problem, which falls below a certain threshold at infinity, already decays exponentially in space, in the sense that v belongs to an exponentially weighted Sobolev space W 1,p. Several extensions of this basic result are presented: to complex-valued systems, to exponential decay in higher order Sobolev spaces and to pointwise estimates. We also prove that every bounded classical solution v of the eigenvalue problemdecays exponentially in space, provided Re λ lies to the right of the essential spectrum. As an application we analyze spinning soliton solutions which occur in the Ginzburg-Landau equation. Our results form the basis for investigating nonlinear stability of rotating waves in higher space dimensions and truncations to bounded domains.

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“…The method not only works for traveling, rotating or meandering waves but also for other relative equilibria and similarity solutions with a more complicated symmetry group, e.g. [3] or [14]. The idea of the method is to write the equation in new (time dependent) coordinates and to split the evolution of the solution into an evolution of the profile and an evolution in a symmetry group which brings the profile into the correct position via the group action.…”

Section: Introductionmentioning

confidence: 99%

An IMEX-RK scheme for capturing similarity solutions in the multidimensional Burgers’s equation

Rottmann-Matthes

2017

IMA Journal of Numerical Analysis

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In this paper we introduce a new, simple and efficient numerical scheme for the implementation of the freezing method for capturing similarity solutions in partial differential equations. The scheme is based on an IMEX-Runge-Kutta approach for a method of lines (semi-)discretization of the freezing partial differential algebraic equation (PDAE). We prove second order convergence for the time discretization at smooth solutions in the ODE-sense and we present numerical experiments that show second order convergence for the full discretization of the PDAE. As an example serves the multi-dimensional Burgers' equation. By considering very different sizes of viscosity, Burgers' equation can be considered as a prototypical example of general coupled hyperbolicparabolic PDEs. Numerical experiments show that our method works perfectly well for all sizes of viscosity, suggesting that the scheme is indeed suitable for capturing similarity solutions in general hyperbolic-parabolic PDEs by direct forward simulation with the freezing method.

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Stability and Computation of Dynamic Patterns in PDEs

Cited by 9 publications

References 56 publications

Computation and stability of waves in equivariant evolution equations

Computation and stability of waves in equivariant evolution equations

Spatial decay of rotating waves in reaction diffusion systems

An IMEX-RK scheme for capturing similarity solutions in the multidimensional Burgers’s equation

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