Uncoupled moving finite elements

An "uncoupled" moving finite element method is developed for solving transient scalar convection-diffusion equations in one dimension. The method is a variant of the moving finite element method proposed by Miller and Miller, but uses local, explicit formulas to determine the evolution of a continuous, piecewise linear approximation to the solution. The computation for one time step is straightforward. Once fluxes are calculated, the motion of each element is determined by conservation equations for both mass and its first moment. Node motion then follows from element motion by considering each node as the intersection of two elements. Integration in time may be explicit. The distribution of nodes obtained by this method is very good if the flux across a node is calculated based on the slope of the secant through its two neighboring nodes. The method is unfortunately subject to a singularity when two successive elements have the same slope. A procedure for regularizing the singularity is proposed that includes as an option the removal and insertion of nodes. While the regularization procedure is successful, it is complex and costly. Further research is needed to realize the potential of this method.;A stabilized explicit integration procedure has also been developed. This procedure allows explicit integration of moderately stiff systems of ordinary differential equations. The Euler algorithm is combined with the conjugate-gradients method. After an Euler step is taken, conjugate-gradients points are used, first, to determine if the Euler procedure is oscillating and, second, if so, to remove the oscillating component from the time derivative so that a stabilized step may be taken.