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We introduce a general framework for the construction of well-balanced finitevolume methods for hyperbolic balance laws. We use the phrase well-balancing ina broader sense, since our proposed method can be applied to exactly follow anysolution of any system of hyperbolic balance laws in multiple spatialdimensions and not only time independent solutions. The solution has to beknown a priori, either as an analytical expression or as discrete data. Theproposed framework modifies the standard finite volume approach such that thewell-balancing property is obtained and in case the method is high orderaccurate, this is maintained under our modification. We present numerical testsfor the compressible Euler equations with and without gravity source term andwith different equations of state, and for the equations of compressible idealmagnetohydrodynamics.

High order well-balanced finite volume methods for multi-dimensional systems of hyperbolic balance laws