4.5 Going further: High-order time schemes

When using spectral methods in time-dependent problems, it is sometimes frustrating to have such accurate numerical techniques for the evaluation of spatial derivatives and the integration of elliptic PDEs, while the time derivatives and hyperbolic PDEs do not benefit from spectral convergence. Some tentative studies are being undertaken in order to represent the time interval by spectral methods as well [113]. In their spherically-symmetric study of the wave equation in Minkowski spacetime, Hennig and Ansorg have applied spectral methods to both spatial and time coordinates. Moreover, they have used a conformal compactification of Minkowski spacetime, making the wave equation singular at null infinity. They have obtained nicely accurate and spectrally convergent solutions, even to a nonlinear wave equation. If these techniques can be applied in general three-dimensional simulations, it would really be a great improvement.

Nevertheless, there are other, more sophisticated and accurate time-integration techniques that are currently being investigated for several stiff PDEs [124Jump To The Next Citation Point], including Korteweg–de Vries and nonlinear Schrödinger equations [129Jump To The Next Citation Point]. Many such PDEs share the properties of being stiff (very different time scales/characteristic frequencies) and combining low-order nonlinear terms with higher-order linear terms. Einstein’s evolution equations can also be written in such a way [37Jump To The Next Citation Point]. Let us consider a PDE

∂u- ∂t = Lu + 𝒩 u, (181 )
using the notation of Section 4.1.1 and 𝒩 as a nonlinear spatial operator. Following the same notation and within spectral approximation, one recovers
∂UN ----- = LN UN + 𝒩N UN . (182 ) ∂t
We will now describe five methods of solving this type of ODE (see also [124Jump To The Next Citation Point]):

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