دانلود رایگان مقاله تقریب انتگرال برای پخش کلاسیکی و هیدرودینامیک ذرات هموار

Abstract

The contribution of the paper is the approximation of a classical diffusion operator by an integral equation with a volume constraint. A particular focus is on classical diffusion problems associated with Neumann boundary conditions. By exploiting this approximation, we can also approximate other quantities such as the flux out of a domain. Our analysis of the model equation on the continuum level is closely related to the recent work on nonlocal diffusion and peridynamic mechanics. In particular, we elucidate the role of a volumetric constraint as an approximation to a classical Neumann boundary condition in the presence of physical boundary. The volume-constrained integral equation then provides the basis for accurate and robust discretization methods. An immediate application is to the understanding and improvement of the Smoothed Particle Hydrodynamics (SPH) method.

6. Conclusion

Smoothed Particle Hydrodynamics is a numerical method for discretizing PDES and its approximation to the classical diffusion problem provides a convenient setting in which to understand the SPH method. There have been many studies of SPH over the years, yet a systematic and rigorous mathematical framework is still lacking. We demonstrate in this work that a suitable “volume-constrained” problem with the integral operator may serve as a conduit between the boundary value problem involving the inhomogeneous diffusion operator and its SPH approximations. This leads to a new mathematical framework based on the recently developed theory for volume-constrained nonlocal diffusion problems and their numerical approximations. The new approach allows us to both distinguish and relate two key limiting processes, namely, the use of an integral operator with a kernel function with a diminishing support to approximate the differential operator and the use of a numerical quadrature with a diminishing mesh size to approximate the integral operator. The new approach also enables the construction of a better-behaved SPH discretization scheme given that the volume constrained continuum problem is well-posed. The modified SPH discretization can therefore be shown to converge to the solution of the volume constrained problem as the number of particles increase. Then by letting the support of the nonlocal kernel decrease to zero, the solution of the volume-constrained problem approximates that of a boundary value problem. It is this two-step process of discretizing the integral operator and letting both the number of points increase and the support of nonlocal kernel decrease that proves critical in understanding carefully the underlying approximations.