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This paper is concerned with the numerical solution of partial differential equations describing fluid flow problems in real space and in phase space. One important goal is to show conclusively that the Accurate Space Derivative methods can be used with success for solving such problems numerically. We describe a method for the numerical solution of the Korteweg-de Vries-Burgers equation. We show numerically that the solution of this equation evolves asymptotically into a steady shock wave with monotonic and oscillatory profile. We present numerical solutions of the Vlasov-Poisson system of equations which describes the motion of an ideal incompressible fluid in phase space. These problems are related to longitudinal oscillations in two- and three-dimensional phase space.

Flow computations with accurate space derivative methods