Matthias Wesenberg

A New Approach to Divergence Cleaning in Magnetohydrodynamic Simulations

In this paper we present a number of approaches for reducing the divergence errors in magnetohydrodynamic simulations. The methods are derived from a general framework, which for example also includes the Hodge projection scheme. The corrections can be easily added to an existing scheme as is demonstrated for a finite-volume scheme. Numerical results in 2d and 3d confirm the advantages of our approach.

Numerical Methods for the Real Gas MHD Equations

In recent years many numerical schemes — mainly based on approximate Riemann solvers — for the equations of ideal magnetohydrodynamics (MHD) have been developed; their robustness and efficiency have been shown in many examples. But since the underlying equation of state (EOS) of an ideal gas is far from reality in many applications (e.g. solar physics), these numerical schemes have to be extended to cope with a more general EOS. For the Euler equations of gas dynamics two general approaches for this extension have recently been proposed. We will show that they can also be applied to MHD. Furthermore we will validate the resulting schemes and will discuss some important aspects of their behaviour.

Efficient Divergence Cleaning in Three-Dimensional MHD Simulations

We present the results of first realistic simulations using our state-of- the-art MHD code on unstructured tetrahedral meshes in 3d. The code incorporates local grid adaption with dynamic load balancing and relies on a recently proposed approximate Riemann solver. We demonstrate that it is absolutely crucial to control the divergence of the magnetic field and that our new hyperbolic divergence cleaning approach works well also in 3d.

Efficient Higher-Order Finite Volume Schemes for (Real Gas) Magnetohydrodynamics

The computational costs for solving the real gas Euler or MHD equations can be strongly reduced if we use an adaptive table instead of the full equation of state. We demonstrate this behaviour for an example from solar physics. Moreover, we introduce a new limiter suitable for second-order finite volume schemes which are based on linear reconstructions on unstructured triangular grids. This new limiter cures several problems of the approaches commonly used. Finally, we show that local grid adaption always seems to pay off in id and 2d, whereas a high-resolution first-order scheme can be more efficient (in terms of computational time versus error) than the second-order schemes.

MHD instabilities arising in solar physics: A numerical approach

Hydrodynamic instabilities are the source of many interesting physical phenomena in fluid dynamics. In this paper we considermagnetohydrodynamic (MHD)instabilities, in particular of Rayleigh-Taylor type. Our numerical studies are motivated by a specific application in solar physics: The development of sun spots — which can be observed from earth — is connected to magnetic field concentrations which develop in the solar convection zone. Driven by magnetic forces, these so-called flux tubes rise through the atmosphere. They are fragmented due to Rayleigh-Taylor type instabilities, and their initially simple structure is perturbed by secondary instabilities of Kelvin-Helmholtz type. An efficient numerical simulation of this complex scenario (large area with small scale structures) requires the incorporation of techniques like local adaptivity and parallelization. At the same time the code must be able to resolve the two basic instabilities in a reliable manner. We focus on these two issues and their interplay.

Godunov-Type Schemes for the MHD Equations

If dissipative effects are neglected, the equations of ideal mag-netohydrodynamics (MHD) are a mathematical model for the flow of a compressible, electrically conducting fluid which is influenced by a magnetic field. They are derived from the Euler equations of fluid dynamics and the Maxwell equations and form a hyperbolic system of conservation laws. Since its behaviour is much more complicated than the Euler system’s, theoretical results and numerical schemes have not yet reached the same level as in the Euler case. This paper focuses on available approximate MHD Riemann solvers, which can be used in Godunov-type finite volume schemes: we present results of an extensive comparison, which justify the choice of the solver we use in our multidimensional code for astrophysical simulations. Moreover, we summarize some important properties of the MHD system and explain how they may influence the solutions’ structure. We conclude with two 2D applications from solar physics.

Comparison of Finite Volume and Discontinuous Galerkin Methods of Higher Order for Systems of Conservation Laws in Multiple Space Dimensions

The methods most frequently used in computational fluid mechanics for solving the compressible Navier-Stokes or compressible Euler equations are finite volume schemes on structured or on unstructured grids. First order as well as higher order space discretizations of MUSCL type, including flux limiters and higher order Runge- Kutta methods for the time discretization, guarantee robust and accurate schemes. But there is an important difficulty. If one increases the order, the stencil for the space discretization increases too, and the scheme becomes very expensive. Therefore schemes with more compact stencils are necessary. Discontinuous Galerkin schemes in the sense of [3] are of this type. They are identical to finite volume schemes in the case of formal first order, and for higher order they use nonconformal ansatz functions whose restrictions to single cells are polynomials of higher order. Therefore they seem to be more efficient and it is of highest interest to compare finite volume and discontinuous Galerkin methods for real applications with respect to their efficiency. Experiences [1] with the Euler equations of gas dynamics indicate that the discontinuous Galerkin methods have some advantages. Since there are no systematic studies available in the literature, we will present in this paper some numerical experiments for hyperbolic conservation laws in multiple space dimensions to compare their efficiency for different situations. As important instances of hyperbolic conservation laws we consider the Euler equations of gas dynamics and Lundquist’s equations of ideal magneto-hydrodynamics (MHD). Furthermore we have found a new limiter which improves the results from [14]. Similar studies have been done in [4].

Absorbing Boundary Conditions for Astrophysical MHD Simulations

Many problems for systems of conservation laws are formulated either on infinite domains or on domains which are by orders of magnitude larger than the interesting structures. In the first case, it is often impossible to find an exact representation of the problem which is suitable for numerical simulations. But even in the second case it can be difficult to perform the simulation on the whole domain, since much computational effort is wasted in uninteresting regions. Therefore the size of the computational domain has to be reduced, which introduces new boundaries without physical meaning. At these artificial boundaries suitable boundary conditions for the PDEs have to be formulated.