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Computations have helped elucidate the dynamics of Earth's mantle for severaldecades already. The numerical methods that underlie these simulations havegreatly evolved within this time span, and today include dynamically changingand adaptively refined meshes, sophisticated and efficient solvers, andparallelization to large clusters of computers. At the same time, many of thesemethods -- discussed in detail in a previous paper in this series -- weredeveloped and tested primarily using model problems that lack many of thecomplexities that are common to the realistic models our community wants tosolve today.  With several years of experience solving complex and realistic models, wehere revisit some of the algorithm designs of the earlier paper and discuss theincorporation of more complex physics. In particular, we re-consider timestepping and mesh refinement algorithms, evaluate approaches to incorporatecompressibility, and discuss dealing with strongly varying materialcoefficients, latent heat, and how to track chemical compositions andheterogeneities. Taken together and implemented in a high-performance,massively parallel code, the techniques discussed in this paper then allow forhigh resolution, 3d, compressible, global mantle convection simulations withphase transitions, strongly temperature dependent viscosity and realisticmaterial properties based on mineral physics data.

High Accuracy Mantle Convection Simulation through Modern Numerical Methods. II: Realistic Models and Problems