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‐order time integrators can deliver the optimal performance of highly accurate and robust spatial discretizations such as isogeometric analysis. The generalized‐α$$ \alpha $$ 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‐α$$ \alpha $$ 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‐α$$ \alpha $$ method in the new higher‐order methods; we increase the accuracy of the generalized‐α$$ \alpha $$ method while keeping the unconditional stability and user‐control features on the high‐frequency numerical dissipation. The methodology solves k>1,k∈ℕ$$ k>1,k\in \mathbb{N} $$ matrix problems and updates the system unknowns, which correspond to higher‐order terms in Taylor expansions to obtain (3/2k)th$$ \left(3/2k\right)\mathrm{th} $$‐order method for even k$$ k $$ and (3/2k+1/2)th$$ \left(3/2k+1/2\right)\mathrm{th} $$‐order for odd k$$ k $$. A single parameter ρ∞$$ {\rho}^{\infty } $$ 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 ρ∞=0$$ {\rho}^{\infty }=0 $$ 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. [ABSTRACT FROM AUTHOR]