In this paper, we consider a class of discontinuous Galerkin (DG) methods for one-dimensional nonlocal diffusion (ND) problems. The nonlocal models, which are integral equations, are widely used in describing many physical phenomena with long-range interactions. The ND problem is the nonlocal analog of the classic diffusion problem, and as the interaction radius (horizon) vanishes, then the nonlocality disappears and the ND problem converges to the classic diffusion problem. Under certain conditions, the exact solution to the ND problem may exhibit discontinuities, setting it apart from the classic diffusion problem. Since the DG method shows its great advantages in resolving problems with discontinuities in computational fluid dynamics over the past several decades, it is natural to adopt the DG method to compute the ND problems. Based on [Du-Ju-Lu-Tian-CAMC2020], we develop the DG methods with different penalty terms, ensuring that the proposed DG methods have local counterparts as the horizon vanishes. This indicates the proposed methods will converge to the existing DG schemes as the horizon vanishes, which is crucial for achieving asymptotic compatibility. Rigorous proofs are provided to demonstrate the stability, error estimates, and asymptotic compatibility of the proposed DG schemes. To observe the effect of the nonlocal diffusion, we also consider the time-dependent convection-diffusion problems with nonlocal diffusion. We conduct several numerical experiments, including accuracy tests and Burgers' equation with nonlocal diffusion, and various horizons are taken to show the good performance of the proposed algorithm and validate the theoretical findings.

翻译：本文研究了一维非局部扩散问题的一类间断伽辽金方法。非局部模型作为积分方程，被广泛用于描述具有长程相互作用的物理现象。非局部扩散问题是经典扩散问题的非局部类比，当相互作用半径趋于零时，非局部性消失，非局部扩散问题收敛于经典扩散问题。在一定条件下，非局部扩散问题的精确解可能出现间断，这使其区别于经典扩散问题。由于过去几十年间，间断伽辽金方法在计算流体力学中处理间断问题方面展现出显著优势，自然可将其应用于非局部扩散问题的计算。基于[Du-Ju-Lu-Tian-CAMC2020]的工作，我们发展了具有不同惩罚项的间断伽辽金方法，确保所提方法在作用半径趋于零时具有对应的局部格式。这表明所提方法将随作用半径消失而收敛于现有间断伽辽金格式，这对于实现渐近相容性至关重要。我们提供了严格的数学证明，以验证所提间断伽辽金格式的稳定性、误差估计及渐近相容性。为观察非局部扩散效应，我们还研究了含非局部扩散项的瞬态对流-扩散问题。通过开展包括精度测试和非局部扩散Burgers方程在内的多组数值实验，并采用不同作用半径进行验证，结果表明所提算法性能良好，理论分析得到有效验证。