Simulation of the Plasma-Sheath Equation on Condensing Grids

We consider the numerical solution of a Tonks–Langmuir integro-differential equation with an Emmert kernel, which describes the behavior of the potential both in the main plasma volume and in the thin boundary (Langmuir) layer. In the volume plasma the potential varies insignificantly, while in the thin layer near the wall (the sheath) it experiences rapid variation. To correctly resolve the behavior of the sheath solution, the numerical region is partitioned into several intervals with a uniform discrete grid in each interval. The interval lengths and the grid increments are successively halved. The second derivative is approximated at the halving points using a nonsymmetrical stencil, which ensures second-order approximation. Near the boundary of the numerical region a step-doubling condensation grid is used, which also ensures second-order approximation of the second-derivative operator. The condensation grid and the numerical algorithm are constructed. Some numerical results are reported.