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We put forward a new type of spectral method for the direct numericalsimulation of flows where anisotropy or very fine boundary layers are present.The mean idea is to take advantage of the fact that such structures aredissipative and that their presence should reduce the number of degrees offreedom of the flow, when paradoxically, their fine resolution incurs extracomputational cost in most current methods. The principle of this method is touse a functional basis with elements that already include these fine structureso as to avoid these extra costs. This leads us to develop an algorithm toimplement a spectral method for arbitrary functional bases, and in particular,non-orthogonal ones. We construct a basic implementation of this algorithm tosimulate Magnetohydrodynamic (MHD) channel flows with an externally imposed,transverse magnetic field, where very thin boundary layers are known to developalong the channel walls. In this case, the sought functional basis can be builtout of the eigenfunctions of the dissipation operator, which incorporate theseboundary layers, and it turns out to be non-orthogonal. We validate this newscheme against numerical simulations of freely decaying MHD turbulence based ona Finite Volume code and it is found to provide accurate results. Its abilityto fully resolve wall-bounded turbulence with a number of modes close to thatrequired by the dynamics is demonstrated on a simple example. This opens theway to full blown simulations of MHD turbulence under very high magneticfields, which were until now too computationally expensive, as thecomputational cost of the proposed method, in contrast to traditional methods,does not depend on the intensity of the magnetic field.

Spectral Direct Numerical Simulations of low Rm MHD channel flows based on the least dissipative modes