Multi-Adaptive Galerkin Methods for ODEs II: Implementation and Applications

Continuing the discussion of the multi-adaptive Galerkin methods mcG(q) and mdG(q) presented in [A. Logg, SIAM J. Sci. Comput., 24 (2003), pp. 1879-1902], we present adaptive algorithms for global error control, iterative solution methods for the discrete equations, features of the implementation Tanganyika, and computational results for a variety of ODEs. Examples include the Lorenz system, the solar system, and a number of time-dependent PDEs

Explicit Time-Stepping for Stiff ODEs

We present a new strategy for solving stiff ODEs with explicit methods. By adaptively taking a small number of stabilizing small explicit time steps when necessary, a stiff ODE system can be stabilized enough to allow for time steps much larger than what is indicated by classical stability analysis. For many stiff problems the cost of the stabilizing small time steps is small, so the improvement is large. We illustrate the technique on a number of well-known stiff test problems

On the zero-stability of multistep methods on smooth nonuniform grids

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid (Formula presented.) can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., (Formula presented.), where (Formula presented.) and the map (Formula presented.) is monotonically increasing with (Formula presented.) and (Formula presented.). The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines (Formula presented.), and a tolerance requirement which determines N. Given any strongly stable multistep method, there is an (Formula presented.) such that the method is zero stable for (Formula presented.), provided that (Formula presented.). Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy (Formula presented.) as (Formula presented.). The results are exemplified for BDF-type methods

Runge–Kutta–Möbius methods

In the numerical integration of nonlinear autonomous initial value problems, the computational process depends on the step size scaled vector field hf as a distinct entity. This paper considers a parameterized transformation hf↦hf∘(I-γhf)-1,and its role in the finite step size stability of singly diagonally implicit Runge—Kutta (SDIRK) methods. For a suitably chosen γ> 0 , the transformed map is Lipschitz continuous with a reasonably small constant, whenever hf is negative monotone. With this transformation, an SDIRK method is equivalent to an explicit Runge–Kutta (ERK) method applied to the transformed vector field. Through this mapping, the SDIRK methods’ A-stability, and linear order conditions are investigated. The latter are closely related to approximations of the exponential function e z that are polynomial in z, when considering ERK methods, and polynomial in terms of the transformed variable z(1 - γz) - 1, in case of SDIRK methods. Considering the second family of methods, and expanding the exponential function in terms of this transformed variable, the coefficients can be expressed in terms of Laguerre polynomials. Lastly, a family of methods is constructed using the transformed vector field, and its order conditions, A-stability, and B-stability are investigated