Range-Discrete Mesh Methods For Convection Problems And Convection-Dominated Diffusion Problems

Convection flow problems and convection-dominated diffusion problems arise in a wide range of important applications. The main feature of such flow problems is that multi-scale structures exist in the field: there are sharp fronts or even shocks where the solution varies drastically in small regions, while the solution varies smoothly and gen-tly outside of these regions. Accurate solutions of the flow field, including the exact distribution of those fronts, are important. However, most traditional numerical meth-ods introduce massive numerical diffusion that over-smooths the fronts; higher order methods may produce oscillations which cause stability problems. To increase accu-racy, the mesh near a front must be refined sufficiently. While using full-field refined grid requires numerous computational resource which may not be tolerated, using adap-tive mesh is practical to gain high resolution solutions. However, general adaptive mesh methods involve complex adaptive strategies and large amount of re-mesh operations,which may increase the computational complexity and introduce numerical diffusion.It is noticed that most numerical methods are based on spatial discrete meshes,and an interpretation of their shortcomings in solving fronts is that,zero-width shocks or nearly zero-width fronts are described by limited small cells with spatial discrete meshes. In the consideration of this reason, a novel discrete mesh named as range-discrete mesh is presented in this paper. The mesh is formed by discretization in the range of the value domain. with range-discrete mesh, shocks can be described sharply,which is essential to reduce numerical diffusion. Moreover, mesh points will automati-cally gather around fronts and are sparse in gently-varied regions. It should be empha-sized that this is an inherent nature and, as a benefit,the adaptivity is achieved along with the solving of flow problems, yet without the need of any re-mesh procedure as traditional adaptive mesh methods do. Furthermore, range-discrete mesh retains simple data structures of regular spatial discrete meshes. It promises a fast and accurate numer-ical method for front propagation flow problems with the application of range-discrete mesh.According to the main feature of range-discrete mesh, the locations of mesh points are tracked to achieve the solution of the flow problems. However, several mathematical difficulties have to be conquered when implementing a range-discrete method.Firstly, range-discrete mesh is applied to solve one-dimension convection flow problems which might develop moving shocks. As the strength of a shock can only be discrete values,a conservation problem might occur if the velocity of a shock is calcu-lated by the well-known Rankine-Hugonoit jump condition. To avoid this, a modified R-H condition is derived. It is found that, with the modified R-H condition, not only conservativeness is guaranteed, but also the location of shocks can be of second order accuracy. Second, as a result of the nonlinearity of the governing equation, mesh points might go ahead of each other, leading to a nonphysical multi-value solution. To solve this problem, a post-processing procedure is designed based on entropy condition and conservation principle. Thus, multi-value problems are avoided and the evolution of shocks can be simulated by the range-discrete method.A Range-discrete mesh method for 1D convection-diffusion problems is devel-oped. Because of the diffusive term in the governing equation, the movements of a mesh point is determined by both its local value and distribution of the solution. How to calculate the velocities of the mesh points is the key to apply the range-discrete mesh.In this paper, a dynamic control volume that are fixed in the value domain while var-ied in the space domain is built around each mesh point. Then,a conservative discrete equation, which determines the velocities of mesh points, can be deduced by control volume integral method on the dynamic volume.For two dimensional water-oil displacement problems, a sequence method which combines the range-discrete method and Finite Analytic Method is proposed. It is found that numerical diffusion of the proposed method is much less than that of FDM. As a benefit, instability of water-oil interface can be observed. For five-spot water injec-tion problems, the differences between the solutions of the proposed method and FDM indicate that instability is one of the internal cause of Grid Orientation Effect.In summary,a range-discrete mesh is proposed and applied preliminarily into nu-merical simulation of some convection and convection-dominated flow problems in this paper. The reliability and effectiveness of range-discrete method is validated by error analysis and numerical tests. It is expected that the range-discrete method offers a fast and accurate tool for the simulation of front propagation flow problems.