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-$\alpha$ 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-$\alpha$ 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 $\rho^\infty$ 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 $\rho^\infty=0$ allows us to obtain an L-stable method. Lastly, we extend this strategy to analyze the accuracy order of a generic method.
翻译:我们提出一个新的高阶时间总体方案, 配有分散用户控制和无条件稳定的全价计算方程式。 高阶时间整合器可以提供高度精确和稳健的空间离散性的最佳性能, 如等离心分析。 通用- 美元法在时间上提供无条件的稳定性和二阶级准确性, 控制离散频谱高频区域中的数字分散性。 我们的目标是扩大通用- 美元方法, 以获得高离心方程式的高度精确和无条件稳定度。 高顺序时间整合器可以提供高度精确和分散的高度空间分解性能。 此外, 我们维持原始的、 第二顺序通用- 美元法的稳定性区域, 也就是提高通用- 美元法的准确性, 同时在高频位数字解析上保持无条件的稳定性和用户控制功能。 方法解决了 $1, krxxxxxxxx 的路径显示 母体问题并更新了我们系统 $ 的更新 美元法 。