An adaptive multiresolution method for ideal magnetohydrodynamics using divergence cleaning with parabolic–hyperbolic correction

•Adaptive multiresolution methods for solving the MHD equations using divergence cleaning yield excellent results for the test cases presented.•Harten's thresholding strategy gives the most efficient results.•The MR computations reduce significantly the CPU time and the memory requirements with respect to a regular grid, while maintaining the precision.

We present an adaptive multiresolution method for the numerical simulation of ideal magnetohydrodynamics in two space dimensions. The discretization uses a finite volume scheme based on a Cartesian mesh and an explicit compact Runge–Kutta scheme for time integration. Harten's cell average multiresolution allows to introduce a locally refined spatial mesh while controlling the error. The incompressibility of the magnetic field is controlled by using a Generalized Lagrangian Multiplier (GLM) approach with a mixed hyperbolic–parabolic correction. Different applications to two-dimensional problems illustrate the properties of the method. For each application CPU time and memory savings are reported and numerical aspects of the method are discussed. The accuracy of the adaptive computations is assessed by comparison with reference solutions computed on a regular fine mesh.