At every instant , one can represent the function by a finite set , composed of its time-dependent spectral coefficients or its values at the collocation points. We denote the spectral approximation to the operator , together with the boundary conditions, if the tau or collocation method is used. is, therefore, represented as an matrix. This is the method of lines, which allows one to reduce a PDE to an ODE, after discretization in all but one dimensions. The advantage is that many ODE integration schemes are known (Runge–Kutta, symplectic integrators, ...) and can be used here. We shall suppose an equally-spaced grid in time, with the timestep noted and .
In order to step from to , one has essentially two possibilities: explicit and implicit schemes. In an explicit scheme, the action of the spatial operator on must be computed to explicitly get the new values of the field (either spatial spectral coefficients or values at collocation points). A simple example is the forward Euler method:implicit scheme one must solve for a boundary value problem in term of at each timestep: it can be performed in the same way as for the solution of the elliptic equation (62) presented in Section 2.5.2. The simplest example is the backward Euler method:
The basic definition of stability for an ODE integration scheme is that, if the timestep is lower than some threshold, then , with constants and independent of the timestep. This is perhaps not the most appropriate definition, since in practice one often deals with bounded functions and an exponential growth in time would not be acceptable. Therefore, an integration scheme is said to be absolutely stable (or asymptotically stable), if remains bounded, . This property depends on a particular value of the product . For each time integration scheme, the region of absolute stability is the set of the complex plane containing all the for which the scheme is absolutely stable.
Finally, a scheme is said to be -stable if its region of absolute stability contains the half complex plane of numbers with negative real part. It is clear that no explicit scheme can be -stable due to the CFL condition. It has been shown by Dahlquist  that there is no linear multistep method of order higher than two, which is -stable. Thus implicit methods are also limited in timestep size, if more than second-order accurate. In addition, Dahlquist  shows that the most accurate second-order -stable scheme is the trapezoidal one (also called Crank–Nicolson, or second-order Adams–Moulton scheme)
Figures 20 and 21 display the absolute stability regions for the Adams–Bashforth and Runge–Kutta families of explicit schemes (see, for instance, ). For a given type of spatial linear operator, the requirement on the timestep usually comes from the largest (in modulus) eigenvalue of the operator. For example, in the case of the advection equation on , with a Dirichlet boundary condition,Second Dahlquist barrier , implicit time marching schemes of order higher than two also have such a limitation.
An important issue in determining the absolute stability of a time-marching scheme for the solution of a given PDE is the computation of the spectrum of the discretized spatial operator (120). As a matter of fact, these eigenvalues are those of the matrix representation of , together with the necessary boundary conditions for the problem to be well posed (e.g., ). If one denotes the number of such boundary conditions, each eigenvalue (here, in the case of the tau method) is defined by the existence of a non-null set of coefficients such that
As an example, let us consider the case of the advection equation (first-order spatial derivative) with a Dirichlet boundary condition, solved with the Chebyshev-tau method (122). Because of the definition of the problem (124), there are “eigenvalues”, which can be computed, after a small transformation, using any standard linear algebra package. For instance, it is possible, making use of the boundary condition, to express the last coefficient as a combination of the other ones
This way of determining the spectrum can be, of course, generalized to any linear spatial operator, for any spectral basis, as well as to the collocation and Galerkin methods. Intuitively from CFL-type limitations, one can see that in the case of the heat equation (), explicit time-integration schemes (or any scheme that is not -stable) will have a severe timestep limitation of the type:
It is sometimes possible to use a combination of implicit and explicit schemes to loosen a timestep restriction of the type (123). Let us consider, as an example, the advection equation with nonconstant velocity on ,)
Living Rev. Relativity 12, (2009), 1
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