4 Time-Dependent Problems

From a relativistic point of view, the time coordinate could be treated in the same way as spatial coordinates and one should be able to achieve spectral accuracy for the time representation of a spacetime function f(t,x,y,z ) and its derivatives. Unfortunately, this does not seem to be the case and we are neither aware of any efficient algorithm dealing with the time coordinate, nor of any published successful code solving any of the PDEs coming from Einstein’s equations, with the recent exception of the 1+1 dimensional study by Hennig and Ansorg [113Jump To The Next Citation Point]. Why is time playing such a special role? It is not easy to find in the literature on spectral methods a complete and comprehensive study. A first standard explanation is the difficulty, in general, of predicting the exact time interval on which one wants to study the time evolution. In addition, time discretization errors in both finite difference techniques and spectral methods are typically much smaller than spatial ones. Finally, one must keep in mind that, contrary to finite difference techniques, spectral methods store all global information about a function over the whole time interval. Therefore, one reason may be that there are strong memory and CPU limitations to fully three-dimensional simulations; it is already very CPU and memory consuming to describe a complete field depending on 3+1 coordinates, even with fewer degrees of freedom, as is the case for spectral methods. But the strongest limitation is the fact that, in the full 3+1 dimensional case, the matrix representing a differential operator would be very big; it would therefore be very time consuming to invert it in a general case, even with iterative methods.

More details on the standard, finite-difference techniques for time discretization are given in Section 4.1. Due to the technical complexity of a general stability analysis, we restrict the discussion of this section to the eigenvalue stability (Section 4.1) with the following approach: the eigenvalues of spatial operator matrices must fall within the stability region of the time-marching scheme. Although this condition is only a necessary one and, in general, is not sufficient, it provides very useful guidelines for selecting time-integration schemes. A discussion of the imposition of boundary conditions in time-dependent problems is given in Section 4.2. Section 4.3 then details the stability analysis of spatial discretization schemes, with the examples of heat and advection equations, before the details of a fully-discrete analysis are given for a simple case (Section 4.4).

 4.1 Time discretization
  4.1.1 Method of lines
  4.1.2 Stability
  4.1.3 Spectrum of simple spatial operators
  4.1.4 Semi-implicit schemes
 4.2 Imposition of boundary conditions
  4.2.1 Strong enforcement
  4.2.2 Penalty approach
 4.3 Discretization in space: stability and convergence
  4.3.1 Lax–Richtmyer theorem
  4.3.2 Energy estimates for stability
  4.3.3 Examples: heat equation and advection equation
 4.4 Fully-discrete analysis
  4.4.1 Strong stability-preserving methods
 4.5 Going further: High-order time schemes

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