## The $$\nabla \cdot B=0$$ constraint in shock-capturing magnetohydrodynamics codes.(English)Zbl 0980.76051

From the summary: Seven schemes to maintain the $$\nabla\cdot {\mathbf B}=0$$ constraint numerically are compared. All these algorithms can be combined with shock-capturing Godunov type base schemes. They fall into three categories: the eight-wave formulation maintains the constraint to truncation error, the projection scheme enforces the constraint in some discretization by projecting the magnetic field, while the five different versions of the constrained transport/central difference type schemes conserve $$\nabla\cdot {\mathbf B}$$ to machine accuracy in some discretization for every grid cell. It is shown that the three constrained transport algorithms can be recast into pure finite volume schemes, and the staggered representation of the magnetic field is unnecessary. Another two new and simple central difference based algorithms are introduced. The properties of the projection scheme are discussed in some detail, and we prove that it has the same order of accuracy as the base scheme, even for discontinuous solutions.

### MSC:

 76M12 Finite volume methods applied to problems in fluid mechanics 76M20 Finite difference methods applied to problems in fluid mechanics 76W05 Magnetohydrodynamics and electrohydrodynamics 76L05 Shock waves and blast waves in fluid mechanics

### Software:

SHASTA; ZEUS; VAC; LASY
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