Math Technical Report Series
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Browsing Math Technical Report Series by Subject "AMS: 35Q30, 41A25, 65M12, 65M60, 76D05"
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Item Achieving High-Order Convergence Rates with Deforming Basis Functions(Department of Mathematical Sciences, 2003) Rossi, Louis F.This article studies the use of moving, deforming elliptical Gaussian basis functions to compute the evolution of passive scalar quantities in a two-dimensional, incompressible flow field. We compute an evolution equation for the velocity, rotation, extension and deformation of the com- putational elements as a function of flow quantities. We find that if one uses the physical flow velocity data calculated from the basis function centroid, the method has only second order spatial accuracy. However, by computing the residual of the numerical method, we can determine adjustments to the centroid data so that the scheme will achieve fourth-order spatial accuracy. Simulations with nontrivial flow parameters demonstrate that the methods exhibit the properties predicted by theory.Item A Comparative Study of Lagrangian Methods Using Axisymmetric and Deforming Blobs(Department of Mathematics, 2003) Rossi, Louis F.This paper presents results from a head-to-head comparison of two Lagrangian methods for solutions to the two-dimensional, incompressible convection-diffusion equations. The first Lagrangian method is an axisymmetric core spreading method using Gaussian basis functions. The second method uses deforming elliptical Gaussian basis functions. Previous results show that the first method has second-order spatial accuracy and the second method has fourth-order spatial accuracy. However, the deforming basis functions require more computational effort per element, so this paper examines computational performance as well as overall accuracy. The test problem is the deformation and diffusion of ellipsoidal distribution of scalar with an underlying flow field that has closed circular streamlines. The test suite includes moderate, high and infinite Peclet number problems. The results indicate that the performance tradeoff for the sample flow calculation occur at modest problem sizes, and that the fourth-order method offers distinct advantages as a general approach for challenging problems.