抛物型问题的高阶广义$α$方法
Higher-order generalized-$α$ methods for parabolic problems
摘要：
提出一类新的求解抛物型方程的具有耗散用户控制和无条件稳定性的高阶时间推进格式。高阶时间积分器可以提供高精度和鲁棒的空间离散化（如等几何分析）的最佳性能。广义- $α$方法在时间上具有无条件稳定性和二阶精度，并控制离散谱高频区域的数值耗散。我们的目标是扩展广义- $alpha$方法，以获得一个高精度和耗散的离散高频范围内的高阶时间推进方法。此外，我们保持了原来的，二阶广义$alpha$方法的新的高阶方法的稳定区域。也就是说，我们提高了广义$α$方法的精度，同时保持了高频数值耗散的无条件稳定性和用户控制功能。该方法求解$k>1, k\in \mathbb{N}$矩阵问题，并更新系统未知数，这些未知数对应于Taylor展开式中的高阶项，从而得到了$(3/2k)^{th}$阶方法（对于偶数$k$）和$(3/2k+1/2)^{th}$阶方法（对于奇数$k$）.一个单一的参数$ρ^\infty$控制的耗散，和更新程序如下制定的原始二阶方法。此外，我们证明了我们的方法是A-稳定的，设置$ρ^\infty=0$可以让我们得到一个L-稳定的方法。最后，我们将此策略扩展到分析一般方法的精度阶。
英文摘要：
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 $k>1, k\in \mathbb{N}$ 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 $ρ^\infty$ 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 $ρ^\infty=0$ 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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