Implementing lowest-order methods for diffusive problems with a DSEL

Abstract

Industrial simulation software has to manage: (1) the complexity of the underlying physical models, (2) the complexity of numerical methods used to solve the PDE systems, and finally (3) the complexity of the low-level computer science services required to have efficient software on modern hardware. Nowadays, some frameworks offer a number of advanced tools to deal with the complexity related to parallelism in a transparent way. However, high-level complexity related to discretization methods and physical models lacks tools to help physicists to develop complex applications. Generative programming and domain-specific languages (DSL) are key technologies allowing to write code with a high-level expressive language and take advantage of the efficiency of generated code for low-level services. Their application to scientific computing has been up to now limited to finite element (FE) methods and Galerkin methods, for which a unified mathematical framework has been existing for a long time (see projects like Freefem++, Getdp, Getfem++, Sundance, Feel++ Aavatsmark et al. (SIAM J. Sci. Comput., 19:1700-1716, 1998), Fenics project). In reservoir and basin modeling, lowest-order methods are promising methods allowing to handle general meshes. Extending finite volume (FV) methods, Aavatsmark, Barkve, Bøe, and Mannseth propose consistent schemes for nonorthogonal meshes while stability problems are solved with the mimetic finite difference method (MFD) and the mixed/hybrid finite volume methods (MHFV) (Aavatsmark et al. Discretization on non-orthogonal, curvilinear grids for multi-phase flow. In: Christie, M.A., Farmer, C.L., Guillon, O., Heinmann, Z.E. (eds.) Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Norway, 1994). However, the lack of a unified mathematical frame was a serious limit to the extension all of these methods to a large variety of problems. In Aavatsmark et al. (J. Comput. Phys., 127:2-14, 1996), the authors propose a unified way to express FV multipoint scheme and DFM/VFMH methods. This mathematical frame allows us to extend the DSL used for FE and Galerkin methods to lowest-order methods. We focus then on the capability of such language to allow the description and the resolution of various and complex problems with different lowest-order methods. We validate the design of the DSL that we have embedded in C++, on the implementation of several academic problems. We present some convergence results and compare the performance of their implementation with the DSEL to their hand-written counterpart.