### 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 [124], including Korteweg–de Vries and nonlinear
Schrödinger equations [129]. 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 [37]. Let us consider a PDE

using the notation of Section 4.1.1 and as a nonlinear spatial operator. Following the same notation
and within spectral approximation, one recovers
We will now describe five methods of solving this type of ODE (see also [124]):
- Implicit-explicit techniques use an explicit multistep scheme to advance the nonlinear part
, and an implicit one for the linear part.
- Split-step techniques are effective when the equation splits into two equations, which
can be directly integrated (see [129] for examples with the nonlinear Schrödinger and
Korteweg–de Vries equations).
- Integrating factor is a change of variable that allows for the exact solution of the linear part
and explicit multistep method for the integration of the new nonlinear part
- Sliders can be seen as an extension of the implicit-explicit method described above. In addition to
splitting the problem into a linear and nonlinear part, the linear part itself is split into two or three
regions (in Fourier space), depending on the wavenumber. Then, different numerical schemes are used
for different groups of wavenumbers: implicit schemes for high wavenumbers and explicit high-order
methods for low wavenumbers. This method is restricted to Fourier spectral techniques in
space.
- Exponential time-differencing techniques have been known for some time in computational
electrodynamics. These methods are similar to the integrating factor technique, but one considers the
exact equation over one timestep
Various orders for these schemes come from the approximation order of the integral. For example
Kassam and Trefethen [124] consider a fourth-order Runge–Kutta type approximation to this integral,
where the difficulty comes from the accurate computation of functions, that suffer from cancellation
errors.