Some properties of the M3D- C₁ form of the three-dimensional magnetohydrodynamics equations

A set of scalar variables and projection operators for the vector momentum and magnetic field evolution equations is presented that has several unique and desirable properties, making it a preferred system for solving the magnetohydrodynamic equations in a torus with a strong toroidal magnetic field. A "weak form" of these equations is derived that explicitly conserves energy and is suitable for a Galerkin finite element formulation provided the basis elements have C₁ continuity. Systems of reduced equations are discussed, along with their energy conservation properties. An implicit time advance is presented that adds diagonally dominant self-adjoint energy terms to the mass matrix to obtain numerical stability.