Variational Extrapolation of Implicit Schemes for General Gradient Flows
Variational Extrapolation of Implicit Schemes for General Gradient Flows
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DOI：
10.1137/19m1283963
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发表时间：
2019-08
期刊：
影响因子：
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通讯作者：
Alexander Zaitzeff;S. Esedoglu;K. Garikipati
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文献类型：
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作者：
Alexander Zaitzeff;S. Esedoglu;K. Garikipati
我们在非常一般的环境中引入了一类无条件能量稳定、高阶精确的梯度流方案。新方案是最小化运动方法的高阶模拟，用于通过解决一系列优化问题来生成梯度流的时间离散近似。特别是，每个步骤都需要最小化梯度流的相关能量加上运动限制项，即，在相对于内积的最速下降的经典背景下，简单地是二次的。各种现有的无条件稳定数值方法可以被认为是（通常只是时间上精确的一阶）最小化其相关演化方程的运动方案，已经需要在每个时间步长加上二次项的能量优化。因此，我们的方法提供了一种轻松的方法，可以将它们扩展到时间方案中的高阶精确度，同时保持它们的无条件稳定性。从这个意义上说，它可以被视为理查森外推法的变分模拟。
We introduce a class of unconditionally energy stable, high order accurate schemes for gradient flows in a very general setting. The new schemes are a high order analogue of the minimizing movements approach for generating a time discrete approximation to a gradient flow by solving a sequence of optimization problems. In particular, each step entails minimizing the associated energy of the gradient flow plus a movement limiter term that is, in the classical context of steepest descent with respect to an inner product, simply quadratic. A variety of existing unconditionally stable numerical methods can be recognized as (typically just first order accurate in time) minimizing movement schemes for their associated evolution equations, already requiring the optimization of the energy plus a quadratic term at every time step. Therefore, our approach gives a painless way to extend these to high order accurate in time schemes while maintaining their unconditional stability. In this sense, it can be viewed as a variational analogue of Richardson extrapolation.