A finite volume differencing scheme that exactly preserves div(b)=0 on a quadrilateral mesh

Summary form only given, as follows. Many different techniques are used to preserve the div(B)=0 constraint in shock-capturing magnetohydrodynamic codes. Most involve making periodic corrections to the magnetic field to project out the diverging component. Some preserve div(B)=0 exactly, but only on a simple rectangular mesh. The paper presents a new finite volume differencing scheme that is capable of exactly preserving div(B)=0 on a 2D mesh of arbitrarily shaped quadrilateral cells. It has been implemented in MACH2 and used to simulate the formation of field reversed configurations and used to simulate their implosion by current driven liners. In the standard finite volume method, numerical approximations for the differential operators div, curl and grad are constructed from surface integrals over the boundary of a control volume around each field point. The new method uses the same control volume, but only for the divergence. The curl is computed from a control surface and the gradient is computed from a control line. If the control surfaces are the boundaries of the control volume, and the control lines are the boundaries of the control surfaces, then the resulting numerical operators will exactly satisfy div(curl)=0 and curl(grad)=0. The magnetic field is computed from the curl of the electric field (Faraday´s law), so it has zero divergence, which is related to conservation of flux. The new operators also satisfy several adjoint relationships, which are related to other conserved quantities such as energy. Numerical expressions for the new operators are derived and explained and proofs are given for all the exact relationships. Extension of this method to three dimensions is currently under investigation and some preliminary 3D results may be shown if time permits.