热方程、线性输运方程与欧拉刚体方程中的隐藏凸性：一种计算方法
Hidden convexity in the heat, linear transport, and Euler's rigid body equations: A computational approach
AI总结：
本文提出基于有限元的计算方案，验证基于对偶的变分方法可统一求解热方程、输运方程及欧拉刚体方程等不同类型的初边值问题，揭示其隐藏凸性与规范不变性，证明非因果对偶问题可正确近似因果原演化问题。
摘要：
本文开发并采用了一种基于有限元的计算方案，用于评估一种基于对偶的变分方法，以求解一维空间含时的线性热方程与输运偏微分方程，以及描述刚体绕定点转动的欧拉非线性常微分方程组。该公式将初（边）值问题转化为（空间）时间域内的退化椭圆边值问题，这些边值问题对应上述各类问题中经适当设计的对偶泛函的欧拉-拉格朗日方程。我们证明，通过适用于变分公式的统一对偶策略，能够较好地近似求解这一系列抛物型、双曲型及常微分原问题，其中既包含能量耗散问题也包含能量守恒问题。该方案自然地将一族对偶解与唯一的原解关联起来；这种“规范不变性”在我们计算得到的热方程与输运方程的解中得到了验证，包括热方程稳态原解对应瞬态对偶解的情形。研究表明，具有因果性的原演化问题可通过非因果对偶问题得到正确近似。
英文摘要：
A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of Euler for the rotation of a rigid body about a fixed point. The formulation turns initial-(boundary) value problems into degenerate elliptic boundary value problems in (space)-time domains representing the Euler-Lagrange equations of suitably designed dual functionals in each of the above problems. We demonstrate reasonable success in approximating solutions of this range of parabolic, hyperbolic, and ODE primal problems, which includes energy dissipation as well as conservation, by a unified dual strategy lending itself to a variational formulation. The scheme naturally associates a family of dual solutions to a unique primal solution; such `gauge invariance' is demonstrated in our computed solutions of the heat and transport equations, including the case of a transient dual solution corresponding to a steady primal solution of the heat equation. Primal evolution problems with causality are shown to be correctly approximated by non-causal dual problems.
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