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.

中文翻译：

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一般梯度流隐式方案的变分外推

我们在非常一般的设置中为梯度流引入了一类无条件能量稳定的高阶精确方案。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，每一步都需要最小化梯度流的相关能量，加上一个运动限制项，在相对于内积的最陡下降的经典上下文中，它只是二次的。各种现有的无条件稳定数值方法可以被认为是（通常只是一阶准确的时间）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优化能量加上二次项。所以，我们的方法提供了一种轻松的方法，可以将这些扩展到高阶准确的时间方案，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的变分类比。