Symmetry boundary conditions

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Abstract

A simple approach to energy conserving boundary conditions using exact symmetries is described which is especially useful for numerical simulations using the finite difference method. Each field in the simulation is normally either symmetric (even) or antisymmetric (odd) with respect to the simulation boundary. Another possible boundary condition is an antisymmetric perturbation about a nonzero value. One of the most powerful aspects of this approach is that it can be easily implemented in curvilinear coordinates by making the scale factors of the coordinate transformation symmetric about the boundaries. The method is demonstrated for magnetohydrodynamics (MHD), reduced MHD, and a hybrid code with particle ions and fluid electrons. These boundary conditions yield exact energy conservation in the limit of infinite time and space resolution. Also discussed is the interpretation that the particle charge reverses sign at a conducting boundary with boundary normal perpendicular to the background magnetic field.

Introduction

A particular set of boundary conditions can have an important effect on the results of plasma simulations [11], [3], [25], [26]. An important class of boundary conditions is that of energy conserving boundary conditions. Energy conserving boundary conditions are sometimes appropriate for laboratory plasmas [2]. Energy conserving boundary conditions are not ideal for all problems. For instance, in simulation of space plasmas, it is often desirable for energy to radiate out of the system simulating an open boundary [12], [14], [5]. But it is difficult to find boundary conditions that let all of the various wave modes escape out of the simulation region, and such boundary conditions are often much less stable than energy conserving boundary conditions. Typically when numerical codes “blow up” (that is, values of quantities get very large), it happens in such a way that the computational energy grows. Related to this, the most stable boundary conditions are typically energy conserving. For some applications, such as the study of Alfvén waves that are guided along the magnetic field, the boundary condition across the magnetic field should not matter, and an energy conserving boundary condition may be convenient. Finally, energy conservation is one of the best tests of a numerical code, and energy conserving boundary conditions make it easier to check that the total simulation energy is constant.

Here, we demonstrate a simple method to determine and implement energy conserving boundary conditions using symmetries about the boundaries. The boundary conditions appropriate for Cartesian coordinates can be used for curvilinear coordinates if the scale factors describing those coordinates are taken to be symmetric about the boundary. We illustrate our method for several different sets of plasma physics equations. While not all aspects of the boundary conditions we describe here are new, we believe that our clear explanation of how to implement them (which we have not found elsewhere) will be helpful to researchers using fluid or particle simulations.

For the sake of completeness, we have included in Section 5 a description of symmetry boundary conditions for reduced MHD. Much of this material can be found in Ref. [9]. Here we have a more complete description of the boundary conditions for all fields (Table 4) using a notation consistent with that used in the rest of this paper.

Section snippets

Symmetry basics

In Table 1, we list the symmetries used in this paper. We use (+) to indicate that a field value is symmetric relative to a particular boundary, and (−) to indicate that a field value is antisymmetric relative to that boundary. “Symmetric” here means that the field is even across the boundary; that is, f(x0+δx)=f(x0-δx), where x is the coordinate normal to the boundary at x=x0, and δx=x-x0. In this case, the first derivative of f is zero at the boundary. Antisymmetry means that the field is odd

Energy conserving boundary conditions for MHD

The normalized MHD equations areρdvdt=-p+J×B,ρt+·(ρv)=0,pt+γ·(pv)+(1-γ)v·p=(γ-1)ηJ2,J=×B,Bt=-×E,E=-v×B+ηJ,where B is normalized to the background magnetic field B0,ρ is normalized to the background mass density ρ0,v is normalized to the background Alfvén speed VA=B0/μ0ρ0, where μ0 is the permeability of free space, and p is normalized to ρ0VA2.

The first step toward deriving energy conserving boundary conditions is to derive an energy equation. This is found by dotting (24) with v,

Energy conserving boundary conditions for a hybrid code

Normalized hybrid code equations for two species, particle protons (p) and fluid electrons (e) [27], are written asdvmdt=Em-ηJm+vm×Bm,dxmdt=vm,nei=npi=WVimS(xi-xm),Jpi=WVimvmS(xi-xm),Ji=×Bi,vei=Qnei(Jpi-Ji),,peit=-γ·(veipei)-(1-γ)vei·pei+(γ-1)ηJi2,Bit=-×Ei,Ei=-vei×Bi+ηJi,where m is the particle index, i is the grid index (possibly representing grid points on a multidimensional grid), the particle weight W=iVi/Nm,Vi is the volume of grid cell i, and Nm is the number of particles. The

Energy conserving boundary conditions for linear reduced MHD

Now we consider a simple set of linear reduced MHD (RMHD) equationsAt=-b0·ϕ,J=-1B0·AB0B02,·1VA2ϕt=B0·J|B0,where the electromagnetic fields areB=×(b0A)=AB0×B0,E=-ϕ-b0At,and the equations have been written in a form valid for curvilinear coordinates with b0=B0/B0=e1, but assuming J0=×B0=0. The and subscripts indicate components respectively parallel and perpendicular to the background magnetic field B0. We define L and L, the parallel and perpendicular scale

Summary and discussion

We have shown how to implement energy conserving boundary conditions using exact symmetries at the boundaries. This method can be used even for a curvilinear coordinate system if the symmetries are exact. (When equations do not have exact symmetries, it is usually better to let the equilibrium quantities and scale factors be continuous at the boundary.) We demonstrated how to work out the symmetries of the various fields for MHD, a hybrid code, and for linear reduced MHD.

In Section 2, we

Acknowledgments

Work at Dartmouth was supported by NSF grants ATM-0503371 and ATM-0120950 (Center for Integrated Space Weather Modeling, CISM, funded by the NSF Science and Technology Centers Program) and NASA grant NNX08AI36G (Heliophysics Theory Program). RED especially thanks James F. Drake for introducing him to symmetry boundary conditions. We also thank John Lyon, Jan Hesthaven, and Alain Brizard for helpful discussions.

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