Anne Kværnø

Sverre Anmarkrud, Kristian Debrabant, Anne Kværnø

In this paper stochastic partitioned Runge-Kutta (SPRK) methods are considered. A general order theory for SPRK methods based on stochastic B-series and multicolored, multishaped rooted trees is developed. The theory is applied to prove the order of some known methods, and it is shown how the number of order conditions can be reduced in some special cases, especially that the conditions for preserving quadratic invariants can be used as simplifying assumptions.

Kristian Debrabant, Anne Kværnø

For a particular class of Stratonovich SDE problems, here denoted as single integrand SDEs, we prove that by applying a deterministic Runge-Kutta method of order $p_d$ we obtain methods converging in the mean-square and weak sense with order $\lfloor p_d/2\rfloor$. The reason is that the B-series of the exact solution and numerical approximation are, due to the single integrand and the usual rules of calculus holding for Stratonovich integration, similar to the ODE case. The only difference is that integration with respect to time is replaced by integration with respect to the measure induced by the single integrand SDE.

Kristian Debrabant, Anne Kværnø

For stochastic implicit Taylor methods that use an iterative scheme to compute their numerical solution, stochastic B--series and corresponding growth functions are constructed. From these, convergence results based on the order of the underlying Taylor method, the choice of the iteration method, the predictor and the number of iterations, for Itô and Stratonovich SDEs, and for weak as well as strong convergence are derived. As special case, also the application of Taylor methods to ODEs is considered. The theory is supported by numerical experiments.

Alemayehu Adugna Arara, Kristian Debrabant, Anne Kværnø

We discuss stochastic differential equations with a stiff linear part and their approximation by stochastic exponential integrators. Representing the exact and approximate solutions using B-series and rooted trees, we derive the order conditions for stochastic exponential integrators. The resulting general order theory covers both Itô and Stratonovich integration.

Asif Mushtaq, Anne Kværnø, Kåre Olaussen

We show how the standard (St{ö}rmer-Verlet) splitting method for differential equations of Hamiltonian mechanics (with accuracy of order $τ^2$ for a timestep of length $τ$) can be improved in a systematic manner without using the composition method. We give the explicit expressions which increase the accuracy to order $τ^8$, and demonstrate that the method work on a simple anharmonic oscillator.

Asif Mushtaq, Anne Kværnø, Kåre Olaussen

We discuss systematic extensions of the standard (St{ö}rmer-Verlet) splitting method for differential equations of Hamiltonian mechanics, with relative accuracy of order $τ^2$ for a timestep of length $τ$, to higher orders in $τ$. We present some splitting schemes, with all intermediate timesteps real and positive, which increase the relative accuracy to order $τ^{N}$ (for N=4, 6, and 8) for a large class of Hamiltonian systems.

Kristian Debrabant, Anne Kværnø

In this article, we construct a representation formula for stochastic B-series evaluated in a B-series. This formula is used to give for the first time the order conditions of implicit Taylor methods in terms of rooted trees. Finally, as an example we apply these order conditions to derive in a simple manner a family of strong order 1.5 Taylor methods applicable to Itô SDEs.