Boundary Source Terms Research Articles

Stationary Boltzmann's equation with Maxwell's boundary conditions in a bounded domain

AbstractThe paper deals with the stationary Boltzmann equation in a bounded convex domain Ω. The boundary ∂Ω is assumed to be a piecewise algebraic variety of the C2‐class that fulfils Liapunov's conditions. On the boundary we impose the so‐called Maxwell boundary conditions, that is a convex combination of specular and diffusive reflections. The non‐linear Boltzmann equation is considered with additional volume and boundary source terms and it has been proved that for sufficiently small sources the problem possesses a unique solution in a properly chosen subspace of C(Ω × ℝ3). The proof is a refined version of the proof delivered by Guiraud for purely diffusive reflection.

Finite element method for subsurface hydrology using a mixed explicit‐implicit scheme

The mixed explicit‐implicit Galerkin finite element method developed previously by the authors is shown to be ideally suited for a wide class of problems arising in subsurface hydrology. These problems include confined saturated flow, unconfined flow under free surface conditions subject to the Dupuit assumption, flow in aquifers which are partly confined and partly unconfined, axisymmetric flow to a well with storage, and flow in saturated‐unsaturated soils. A single computer program, entitled Flump, can now handle all of these problems. The mixed explicit‐implicit solution strategy employed in the program insures a high level of accuracy and computation efficiency in most cases. It eliminates many of the difficulties that groundwater hydrologists have been encountering in trying to simulate extensive aquifer systems by finite elements. Some of the outstanding features of this solution strategy include an automatic control of time step size, reclassification of nodes from explicit to implicit during execution, automatic adjustment of the implicit time‐weighting factor, and the treatment of boundary conditions and source terms as arbitrary functions of time of the state of the system. Five examples are presented to demonstrate the versatility and power of this new approach. A purely physical derivation of the finite element equations which does not rely on the Galerkin formalism is also included in one of the appendices.

Atmospheric Diffusion and Deposition of Particulate Matter

The physical situation considered is described by a nonhomogeneous, gravity-modified, dimensional, time-varying, convective-diffusive transport equation. This partial differential equation, subject to a nonhomogeneous boundary condition and source term, was solved analytically by integral transforms for a third type (Monin) boundary condition which describes the total flux of particles at the earth boundary. By applying the transform kernels provided, the analytical solutions to the cases of first (Dirichlet) or second (Neumann) type boundary conditions may be obtained directly from the general, modular solution presented. The inclusion of the particle flux boundary condition is a significant improvement over previous three-dimensional studies which either assumed infinite geometry in all three dimensions thus avoiding complicating boundary conditions or used “tilting plume” techniques (corrections applied to gaseous no-deposit solutions) which do not properly represent the concurrent processes of diffusive spread and settling.