Higher-order generalized-α methods for parabolic problems

We propose a new class of high-order time-marching schemes with dissipation control and unconditional stability for parabolic equations. High-ordertime 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 is an element of N matrix problems and updates the system unknowns, which correspond to higher-order terms in Taylor expansions to obtain(3/2k)th-order method for even k and (3/2k+1/2)th-order foroddk. A single parameter rho(infinity) 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(infinity)=0 we obtainan 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 com-pare the method to Runge-Kutta techniques in simulating the Lorenz system.