Multidimensional Least Squares Fluctuation Distribution Schemes with Adaptive Mesh Movement for Steady Hyperbolic Equations

Optimal meshes and solutions for steady conservation laws and systems within a finite volume fluctuation distribution framework are obtained by least squares methods incorporating mesh movement. The problem of spurious modes is alleviated through adaptive mesh movement, the least squares minimization giving an obvious way of determining the movement of the nodes and also providing a link with equidistribution. The iterations are carried out locally node by node, which yields good control of the moving mesh. For scalar equations an iteration which respects the flow of information in the problem significantly accelerates the convergence. The method is demonstrated on a scalar advection problem and a shallow water channel flow problem. For discontinuous solutions we introduce a least squares shock fitting approach which greatly improves the treatment of discontinuities at little extra expense by using degenerate triangles and moving the nodes. Examples are shown for a discontinuous shallow water channel flow and a shocked flow in gasdynamics governed by the compressible Euler equations.