The Discrete Operator Approach to the Numerical Solution of Partial Differential Equations

The design of robust computational physics codes has always been a challenge to application programmers. One of the key di culties in writing multiphysics codes stems from the ine cient handling of spatial discretization and field operations. For example, in the context of the finite volume (FV) method, one often deals with structured and unstructured meshes that are either staggered or collocated. To handle this array of options, a complex data structure is required to represent arbitrary meshes. For structured grids, this approach imposes an unnecessary computational overhead that increases significantly with problem size. Instead, the programmer must write specific components to handle structured meshes thus defeating the purpose of code portability. Furthermore, dedicated discretization schemes are required for di erent types of meshes, thus increasing software complexity. Although modern computational codes rely extensively on a variety of libraries for linear algebra operations, they lack a framework that provides proper application-independent discretization tools. In this manuscript, we present a novel computational paradigm that separates the spatial discretization and field operations from the physics. This paradigm is based on the abstraction of the mathematical operators describing physical processes. An operator corresponds to a precise mathematical object that performs a certain calculation on a field. A field corresponds to any scalar or vector variable required in the solution process. In our model, an operator is represented discretely by a sparse matrix while a field is represented by a vector. At the outset, the discretization process corresponds to a sparse-matrix-vector multiplication. This approach completely decouples the physics from the spatial and field operations thus providing an avenue for improved code design and usability.