Higher-order generalized-α methods for parabolic problems

Abstract

We propose a new class of high-order time-marching schemes with dissipation control and unconditional stability for parabolic equations. High-order time integrators can deliver the optimal performance of highly accurate and robust spatial discretizations such as isogeometric analysis. The generalized-$\alpha$ method delivers unconditional stability and second-order accuracy in time and controls the numerical dissipation in the discrete spectrum's high-frequency region. We extend the generalized-$\alpha$ methodology to obtain high-order time marching methods with high accuracy and dissipation control in the discrete high-frequency range. Furthermore, we maintain the original stability region of the second-order generalized-$\alpha$ method in the new higher-order methods; we increase the accuracy of the generalized-$\alpha$ method while keeping the unconditional stability and user-control features on the high-frequency numerical dissipation. The methodology solves $k>1,k\in \mathbb{N}$ matrix problems and updates the system unknowns, which correspond to higher-order terms in Taylor expansions to obtain $(3/2k)\text{th}$-order method for even $k$ and $(3/2k+1/2)\text{th}$-order for odd $k$. A single parameter ${\rho }^{\infty }$ controls the high-frequency dissipation, while the update procedure follows the formulation of the original second-order method. Additionally, we show that our method is A-stable, and for ${\rho }^{\infty }=0$ we obtain an L-stable method. Furthermore, we extend this strategy to analyze the accuracy order of a generic method. Lastly, we provide numerical examples that validate our analysis of the method and demonstrate its performance. First, we simulate heat propagation; then, we analyze nonlinear problems, such as the Swift–Hohenberg and Cahn–Hilliard phase-field models. To conclude, we compare the method to Runge–Kutta techniques in simulating the Lorenz system.