Solving The Liouville Equation With The General Boundary Element Method Approach.
Free (open access)
Shi-JunLiao and Song-Ping Zhu
The Liouville equation has many applications in mathematical physics and engineering. It is a challenge even when one attempts to solve it numerically as the non-linearity arising from the non-homogeneous term is very strong. In this paper, we apply the general boundary element method (GBEM) proposed by Liao (1995) to solve the Liouville equation defined on a square computational domain. By choosing appropriate boundary conditions so that we can compare our numerical solutions with a set of very special analytical solutions, we can demonstrate the robustness of the GBEM approach in solving this highly nonlinear partial differenti