Pouria Behnoudfar — Page 2

arXiv2019

High-order generalized-$α$ methods

The generalized-$α$ method encompasses a wide range of time integrators. The method possesses high-frequency dissipation while minimizing unwanted low-frequency dissipation and the numerical dissipation can be controlled by the user. The method is unconditionally stable and is of second-order accuracy in time. We extend the second-order generalized-$α$ method to third-order in time while the numerical dissipation can be controlled in a similar fashion. We establish that the third-order method is unconditionally stable. We discuss a possible path to the generalization to higher order schemes. All these high-order schemes can be easily implemented into programs that already contain the second-order generalized-$α$ method.

math.NA↗

arXiv2018

A variationally separable splitting for the generalized-$α$ method for parabolic equations

We present a variationally separable splitting technique for the generalized-$α$ method for solving parabolic partial differential equations. We develop a technique for a tensor-product mesh which results in a solver with a linear cost with respect to the total number of degrees of freedom in the system for multi-dimensional problems. We consider finite elements and isogeometric analysis for the spatial discretization. The overall method maintains user-controlled high-frequency dissipation while minimizing unwanted low-frequency dissipation. The method has second-order accuracy in time and optimal rates ($h^{p+1}$ in $L^2$ norm and $h^p$ in $L^2$ norm of $\nabla u$) in space. We present the spectrum analysis on the amplification matrix to establish that the method is unconditionally stable. Various numerical examples illustrate the performance of the overall methodology and show the optimal approximation accuracy.

math.NA↗