Accurate efficient and scalable computational methods are highly desirable for large-scale scientific computing applications especially for problems exhibiting spatial and temporal multi-resolution scales non-trivial geometries and complex boundary conditions (BSc). For global magnetohydrodynamics (MHD) modelling of space-physics problemshigh-performance approaches could significantly reduce the grid requirements to achieve targeted solution accuracies thereby enabling more affordable yet accurate predictions of space-plasma flows. Key challenges encountered relate to providing solenoidal magnetic fields accurate discretizations on spherical domains capturing of MHD shocks and implementing accurate BCs. This talk gives an overview of a fourth-order finite-volume discretization procedure in combination with a parallel solution-adaptive algorithm for the computation of MHD space plasmas on cubed-sphere grids. Numerical results to demonstrate the accuracy and capability of the multidimensional high-order solution-adaptive cubed-sphere computational framework are presented.