Bifurcation of Homoclinic Structures.

1997 ◽  
Vol 07 (03) ◽  
pp. 527-549
Author(s):  
Jürgen Gerling ◽  
Hartmut Jürgens ◽  
Heinz-Otto Peitgen
Keyword(s):  
Finite Elements ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Periodic Structure ◽  
Nonlinear Boundary ◽  
Specific Area ◽  
Finite Difference Approximation ◽  
Approximation Schemes ◽  
Difference Approximation ◽  
Area Preserving

This paper is a continuation of Gerling et al. [1997] and investigates the periodic structure of a specific area-preserving homeomorphism which is generated by the finite difference approximation for a nonlinear boundary value problem. Moreover, these and prior results are extended to further approximation schemes like collocation and finite elements.

Bifurcation of Homoclinic Structures: I. Finite Difference Approximation and Mechanisms for Spurious Solutions

1997 ◽  
Vol 07 (02) ◽  
pp. 287-317
Author(s):  
Jürgen Gerling ◽  
Hartmut Jürgens ◽  
Heinz-Otto Peitgen
Keyword(s):  
Dynamical System ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Discrete Dynamical System ◽  
Finite Difference Approximation ◽  
Nonlinear Boundary Value Problem ◽  
Difference Approximation ◽  
Spurious Solutions

In the present paper we consider the finite difference approximation for a specific nonlinear boundary value problem which generates spurious solutions. Several mechanisms for spurious solutions will be presented. The key connection is that of the homoclinic structure of an adjoint time discrete dynamical system.

scholarly journals Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

scholarly journals On the finite difference approximation for a parabolic blow-up problem

2007 ◽  
Vol 24 (2) ◽  
pp. 131-160 ◽  
Author(s):  
C. -H. Cho ◽  
S. Hamada ◽  
H. Okamoto
Keyword(s):  
Finite Difference ◽  
Blow Up ◽  
Finite Difference Approximation ◽  
Difference Approximation
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