The discontinuous Galerkin (DG) method is an established method for computing approximate solutions of partial differential equations in many applications. Unlike continuous finite elements, in DG methods numerical fluxes are used to enforce inter-element conditions, and internal/external physical boundary conditions. For elastic wave propagation in complex media several wave types, including dissipative surface and interface waves, are simultaneously supported. The presence of multiple wave types and different physical phenomena pose a significant challenge for numerical fluxes. When modelling surface or interface waves an incompatibility of the numerical flux with the physical boundary condition leads to numerical artefacts. We present a stable and arbitrary order accurate DG method for elastic waves with a physically motivated numerical flux. Our numerical flux is compatible with all well-posed, internal and external, boundary conditions, including linear and nonlinear frictional constitutive equations for modelling spontaneously propagating shear ruptures in elastic solids and dynamic earthquake rupture processes. By construction our choice of penalty parameters yield an upwind scheme and a discrete energy estimate analogous to the continuous energy estimate. We derive a priori error estimate for the DG method proving optimal convergence to discontinuous and nearly singular exact solutions. The spectral radius of the resulting spatial operator has an upper bound which is independent of the boundary and interface conditions, thus it is suitable for efficient explicit time integration. We present numerical experiments in one and two space dimensions verifying high order accuracy and asymptotic numerical stability. We demonstrate the potential of the method for modelling complex nonlinear frictional problems in elastic solids with 2D dynamically adaptive meshes and non-planar topography with 2D curvilinear elements.

不连续伽辽金 (DG) 方法是在许多应用中计算偏微分方程近似解的既定方法。与连续有限元不同，在 DG 方法中，数值通量用于强制执行元素间条件和内部/外部物理边界条件。对于复杂介质中的弹性波传播，同时支持多种波类型，包括耗散表面波和界面波。多种波类型和不同物理现象的存在对数值通量提出了重大挑战。当模拟表面或界面波时，数值通量与物理边界条件的不兼容会导致数值伪影。我们为具有物理激励的数值通量的弹性波提出了一种稳定且任意阶精确的 DG 方法。我们的数值通量与所有适定的内部和外部边界条件兼容，包括线性和非线性摩擦本构方程，用于对弹性固体中的自发传播剪切破裂和动态地震破裂过程进行建模。通过构造我们对惩罚参数的选择产生了一个迎风方案和一个类似于连续能量估计的离散能量估计。我们推导出 DG 方法的先验误差估计，证明对不连续和几乎奇异的精确解的最佳收敛。所得空间算子的谱半径有一个独立于边界和界面条件的上限，因此适用于有效的显式时间积分。我们在一维和二维空间中进行数值实验，验证高阶精度和渐近数值稳定性。我们展示了该方法在具有 2D 动态自适应网格和具有 2D 曲线元素的非平面地形的弹性固体中建模复杂非线性摩擦问题的潜力。