Higher-order generalized-α methods for parabolic problems
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We propose a new class of high-order time-marching schemes with dissipation user-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 method delivers unconditional stability and second-order accuracy in time and controls the numerical dissipation in the discrete spectrum's high-frequency region. Our goal is to extend the generalized-alpha methodology to obtain a high-order time marching methods with high accuracy and dissipation in the discrete high-frequency range. Furthermore, we maintain the stability region of the original, second-order generalized-alpha method foe the new higher-order methods. That is, we increase the accuracy of the generalized-α method while keeping the unconditional stability and user-control features on the high-frequency numerical dissipation. The methodology solve k>1, k∈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 for odd k. A single parameter ρ^∞ controls the dissipation, and the update procedure follows the formulation of the original second-order method. Additionally, we show that our method is A-stable and setting ρ^∞=0 allows us to obtain an L-stable method. Lastly, we extend this strategy to analyze the accuracy order of a generic method.
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