A numerical comparison between degenerate parabolic and quasilinear hyperbolic models of cell movements under chemotaxis

We consider two models which were both designed to describe the movement of eukaryotic cells responding to chemical signals. Besides a common standard parabolic equation for the diffusion of a chemoattractant, like chemokines or growth factors, the two models differ for the equations describing the movement of cells. The first model is based on a quasilinear hyperbolic system with damping, the other one on a degenerate parabolic equation. The two models have the same stationary solutions, which may contain some regions with vacuum. We first explain in details how to discretize the quasilinear hyperbolic system through an upwinding technique, which uses an adapted reconstruction, which is able to deal with the transitions to vacuum. Then we concentrate on the analysis of asymptotic preserving properties of the scheme towards a discretization of the parabolic equation, obtained in the large time and large damping limit, in order to present a numerical comparison between the asymptotic behavior of these two models. Finally we perform an accurate numerical comparison of the two models in the time asymptotic regime, which shows that the respective solutions have a quite different behavior for large times.Comment: One sentence modified at the end of Section 4, p. 1

The Well-posedness of the Null-Timelike Boundary Problem for Quasilinear Waves

The null-timelike initial-boundary value problem for a hyperbolic system of equations consists of the evolution of data given on an initial characteristic surface and on a timelike worldtube to produce a solution in the exterior of the worldtube. We establish the well-posedness of this problem for the evolution of a quasilinear scalar wave by means of energy estimates. The treatment is given in characteristic coordinates and thus provides a guide for developing stable finite difference algorithms. A new technique underlying the approach has potential application to other characteristic initial-boundary value problems.Comment: Version to appear in Class. Quantum Gra

Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure

Characteristic Evolution and Matching

I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress in characteristic evolution is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativistic stars and to current 3D codes that provide pieces of a binary black hole spacetime. Cauchy codes have now been successful at simulating all aspects of the binary black hole problem inside an artificially constructed outer boundary. A prime application of characteristic evolution is to extend such simulations to null infinity where the waveform from the binary inspiral and merger can be unambiguously computed. This has now been accomplished by Cauchy-characteristic extraction, where data for the characteristic evolution is supplied by Cauchy data on an extraction worldtube inside the artificial outer boundary. The ultimate application of characteristic evolution is to eliminate the role of this outer boundary by constructing a global solution via Cauchy-characteristic matching. Progress in this direction is discussed.Comment: New version to appear in Living Reviews 2012. arXiv admin note: updated version of arXiv:gr-qc/050809

Boundary stabilization of quasilinear hyperbolic systems of balance laws: Exponential decay for small source terms

We investigate the long-time behavior of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition

Dispersive nonlinear geometric optics

We construct infinitely accurate approximate solutions to systems of hyperbolic partial differential equations which model short wavelength dispersive nonlinear phenomena. The principal themes are the following. (1) The natural framework for the study of dispersion is wavelength ϵ solutions of systems of partial differential operators in ϵ∂. The natural ϵ-characteristic equation and ϵ-eikonal equations are not homogeneous. This corresponds exactly to the fact that the speeds of propagation, which are called group velocities, depend on the length of the wave number. (2) The basic dynamic equations are expressed in terms of the operator ϵ∂tt. As a result growth or decay tends to occur at the catastrophic rate ect/ϵect/ϵ. The analysis is limited to conservative or nearly conservative models. (3) If a phase ϕ(xx)/ϵ satisfies the natural ϵ-eikonal equation, the natural harmonic phases, nϕ(x)nϕ(x)/ϵ, generally do not. One needs to impose a coherence hypothesis for the harmonics. (4) In typical examples the set of harmonics which are eikonal is finite. The fact that high harmonics are not eikonal suppresses the wave steepening which is characteristic of quasilinear wave equations. It also explains why a variety of monochromatic models are appropriate in nonlinear settings where harmonics would normally be expected to appear. (5) We study wavelength ϵ solutions of nonlinear equations in ϵ∂ for times OO(1). For a given system, there is a critical exponent pp so that for amplitudes O(ϵp)O(ϵp), one has simultaneously smooth existence for t = O(1)t=O(1), and nonlinear behavior in the principal term of the approximate solutions. This is the amplitude for which the time scale of nonlinear interaction is O(1)O(1). (6) The approximate solutions have residual each of whose derivatives is O(ϵn)O(ϵn) for all nn>0. In addition, we prove that there are exact solutions of the partial differential equations, that is with zero residual, so that the difference between the exact solution and the approximate solutions is infinitely small. This is a stability result for the approximate solutions. © 1997 American Institute of Physics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71258/2/JMAPAQ-38-3-1484-1.pd