Implementing a Mesh-Projection Schemes Using the Technology of Adaptive Mesh Refinement

The adaptive mesh refinement (AMR) is a basic method for development of efficient technologies allowing a numerical solution of various problems of continuum mechanics. Numerical modeling of heat-mass transfer, hydrodynamics, structural/fracture mechanics etc. deals with non-linear strongly coupled processes which significantly differ in spatial and temporal scales. Modeling of multiscale correlated phenomena stimulates application of computational technologies using irregular grids, among which the octree meshes are the most known and widely used. Using the created data for tree-structured meshes we developed the dynamic loading balance algorithm aimed at applications to the cluster type parallel computing systems. The developed tools support functionality necessary to implement various numerical models of continuum mechanics. As an example of possible applications we discuss the constructed family of mesh projective approximations to second-order partial differential equations with a variable tensor-type coefficients. The difference operator of the scheme provides energy conservation and possesses the “self-adjoint” property which is inherent to the original differential operator, e.g. in the case of a heat transfer model. We consider numerical results obtained by solution of some model initial boundary value problems for parabolic equations using the developed AMR technique .