The Space-Time Conservation Element and Solution Element Method: A New High-Resolution and Genuinely Multidimensional Paradigm for Solving Conservation Laws

Abstract

A new high-resolution and genuinely multidimensional numerical method for solving conservation laws is being developed. It was designed to avoid the limitations of the traditional methods and was built from ground zero with extensive physics considerations. Nevertheless, its foundation is mathematically simple enough that one can build from it a coherent, robust, efficient, and accurate numerical framework. Two basic beliefs that set the new method apart from the established methods are at the core of its development. The first belief is that, in order to capture physics more efficiently and realistically, the modeling focus should be placed on the original integral form of the physical conservation laws, rather than the differential form. The latter form follows from the integral form under the additional assumption that the physical solution is smooth, an assumption that is difficult to realize numerically in a region of rapid change, such as a boundary layer or a shock. The second belief is that, with proper modeling of the integral and differential forms themselves, the resulting numerical solution should automatically be consistent with the properties derived from the integral and differential forms, e.g., the jump conditions across a shock and the properties of characteristics. Therefore a much simpler and more robust method can be developed by avoiding the explicit use of the above derived properties. Specifically, to capture physics as fully as possible, the method requires that: (i) space and time be unified and treated as a single entity; (ii) both local and global flux conservation in space and time be enforced; and (iii) a multidimensional scheme be constructed without using the dimensional-splitting approach, such that multidimensional effects and source terms (which are scalars) can be modeled more realistically. To simplify mathematics and broaden its applicability as much as possible, the method attempts to use the simplest logical structures and approximation techniques. Specifically, (i) it uses a staggered space-time mesh such that flux at any interface separating two conservation elements can be evaluated internally in a simpler and more consistent manner, without using a separate flux model; (ii) it avoids the use of many well-established techniques such as Riemann solvers, flux splittings, and monotonicity constraints such that the complications and possibly even the limitations associated with them may be avoided; and (iii) it avoids the use of special techniques that are not applicable to more general problems. Furthermore, triangles in 2D space and tetrahedrons in 3D space are used as the basic building blocks of the spatial meshes, such that the method (i) can be used to construct 2D and 3D nondissipative schemes in a natural manner; and (ii) is compatible with the simplest unstructured meshes. Note that while numerical dissipation is required for shock capturing, it may also result in annihilation of small disturbances such as sound waves and, in the case of flow with a large Reynolds number, may overwhelm physical dissipation. To overcome this difficulty, two different and mutually complementary types of adjustable numerical dissipation are introduced in the present development.