Many differential equations with physical backgrounds are described as gradient systems, which are evolution equations driven by the gradient of some functionals, and such problems have energy conservation or dissipation properties. For numerical computation of gradient systems, numerical schemes that inherit the energy structure of the equation play important roles, which are called structure-preserving. The discrete gradient method is one of the most classical framework of structure-preserving methods, which is at most second order accurate. In this paper, we develop a higher-order structure-preserving numerical method for gradient systems, which includes the discrete gradient method. We reformulate the gradient system as a coupled system and then apply the discontinuous Galerkin time-stepping method. Numerical examples suggests that the order of accuracy of our scheme is $(k+1)$ in general and $(2k+1)$ at nodal times, where $k$ is the degree of polynomials.

翻译：许多具有物理背景的微分方程被描述为梯度系统，这些是由某些泛函的梯度驱动的演化方程，此类问题具有能量守恒或耗散性质。在梯度系统的数值计算中，继承方程能量结构的数值格式（称为保结构格式）发挥着重要作用。离散梯度方法是保结构方法中最经典的理论框架之一，但其精度最高仅为二阶。本文针对梯度系统发展了一种高阶保结构数值方法，该方法涵盖了离散梯度方法。我们将梯度系统重构为耦合系统，进而应用不连续伽辽金时间步进法。数值算例表明，所提格式的精度阶通常为$(k+1)$，在节点时刻可达$(2k+1)$，其中$k$为多项式次数。