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We describe a numerical algorithm to integrate the equations of radiationmagnetohydrodynamics in multidimensions using Godunov methods. This algorithmsolves the radiation moment equations in the mixed frame, without invoking anydiffusion-like approximations. The moment equations are closed using a variableEddington tensor whose components are calculated from a formal solution of thetransfer equation at a large number of angles using the method of shortcharacteristics. We use a comprehensive test suite to verify the algorithm,including convergence tests of radiation-modified linear acoustic andmagnetosonic waves, the structure of radiation modified shocks, andtwo-dimensional tests of photon bubble instability and the ablation of denseclouds by an intense radiation field. These tests cover a very wide range ofregimes, including both optically thick and thin flows, and ratios of theradiation to gas pressure of at least 10^{-4} to 10^{4}. Across most of theparameter space, we find the method is accurate. However, the tests also revealthere are regimes where the method needs improvement, for example when both theradiation pressure and absorption opacity are very large. We suggestmodifications to the algorithm that will improve accuracy in this case. Wediscuss the advantages of this method over those based on flux-limiteddiffusion. In particular, we find the method is not only substantially moreaccurate, but often no more expensive than the diffusion approximation for ourintended applications.

A Godunov Method for Multidimensional Radiation Magnetohydrodynamics based on a variable Eddington tensor