In the paper we discuss the transition from kinetic theory to macroscopic uid equations, where the macroscopic equations are deened as asymptotic limits of a kinetic equation. This relation can be used to derive computationally eecient domain decomposition schemes for the simulation of rareeed gas ows close to the continuum limit. Moreover, we present some basic ideas for the derivation of kine...

Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability distribution over the ODE solution. In contrast to prior work, we construct th...

This report analyzes a partitioned time stepping algorithm, meaning a non-iterative, domain decomposition method, which allows different time steps in the fluid region and the porous region for the fully evolutionary Stokes-Darcy problem. The method presented requires only one, uncoupled Stokes and Darcy sub-physics and sub-domain solve per time step. Under a time step restriction of the form 4...

magnetic fields play an important role in creating, driving, and in the evolution of outflows and jets from protostars and accretion disks. on the other hand, the temperature profile of the accretion disks may also affect the structure of the magnetic field and outflows. in this paper, we use the self-similar method in cylindrical coordinates to investigate the effect of the temperature profile...

We consider the development of implicit-explicit time integration schemes for optimal control problems governed by the Goldstein–Taylor model. In the diffusive scaling this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman-Enskog type limiting procedure. For the class of stiffly accurate implicit–explicit Run...

In this note we propose and analyze an implicit-explicit scheme based on second order strong stability preserving time discretisations. We also present some theoretical and numerical stability results for second order Runge Kutta IMEX schemes.

where y(x) denotes the solution of the differential equation. The idea is to use a quadrature formula to estimate the integral of (1). This requires knowledge of the integrand at specified arguments x¿ in (xo, -To + h)—hence we require the values of y(x) at these arguments. A numerical integration method may be used to estimate y(x) for the required arguments. In this way a numerical integratio...

Numerical integration of ordinary differential equations with some invariants is considered. For such a purpose, certain projection methods have proved its high accuracy and efficiency, but sometimes they can exhibit instability. In this paper, a new, highly efficient projection method is proposed based on explicit Runge–Kutta methods. The key there is to employ the idea of the perturbed colloc...

For Hamiltonian systems with non-canonical structure matrix a new class of numerical integrators is proposed. The methods exactly preserve energy, are invariant with respect to linear transformations, and have arbitrarily high order. Those of optimal order also preserve quadratic Casimir functions. The discussion of the order is based on an interpretation as partitioned Runge–Kutta method with ...

In this work, we derive the order conditions for fourth order time splitting schemes in the case of the 1D Vlasov-Poisson system. Computations to obtain such conditions are motivated by the specific Poisson structure of the Vlasov-Poisson system : this structure is similar to Runge-Kutta-Nyström systems. The obtained conditions are proved to be the same as RKN conditions derived for ODE up to t...