A key numerical difficulty in compressible fluid dynamics is the formation of shock waves. Shock waves feature jump discontinuities in the velocity and density of the fluid and thus preclude the existence of classical solutions to the compressible Euler equations. Weak entropy solutions are commonly defined by viscous regularization, but even small amounts of viscosity can substantially change the long-term behavior of the solution. In this work, we propose the first inviscid regularization of the multidimensional Euler equation based on ideas from semidefinite programming, information geometry, geometric hydrodynamics, and nonlinear elasticity. From a Lagrangian perspective, shock formation in entropy solutions amounts to inelastic collisions of fluid particles. Their trajectories are akin to that of projected gradient descent on a feasible set of non-intersecting paths. We regularize these trajectories by replacing them with solution paths of interior point methods based on log determinantal barrier functions. These paths are geodesic curves with respect to the information geometry induced by the barrier function. Thus, our regularization replaces the Euclidean geometry of trajectories with a suitable information geometry. We extend this idea to infinite families of paths by viewing Euler's equations as a dynamical system on a diffeomorphism manifold. Our regularization embeds this manifold into an information geometric ambient space, equipping it with a geodesically complete geometry. Expressing the resulting Lagrangian equations in Eulerian form, we derive a regularized Euler equation in conservation form. Numerical experiments on one and two-dimensional problems show its promise as a numerical tool. While we focus on the barotropic Euler equations for concreteness and simplicity of exposition, our regularization easily extends to more general Euler and Navier-Stokes-type equations.

翻译：可压缩流体动力学中的一个关键数值难点是激波的形成。激波在流体的速度和密度上呈现跳跃间断，因此排除了可压缩欧拉方程经典解的存在性。弱熵解通常通过粘性正则化来定义，但即使少量的粘性也会显著改变解的长期行为。在本文中，我们提出了基于半定规划、信息几何、几何流体动力学和非线性弹性力学思想的多维欧拉方程的第一个无粘正则化方法。从拉格朗日观点来看，熵解中的激波形成相当于流体粒子的非弹性碰撞。其轨迹类似于在不相交路径可行集上的投影梯度下降轨迹。我们通过将这些轨迹替换为基于对数行列式障碍函数的内点方法的解路径来对其进行正则化。这些路径是关于障碍函数所诱导的信息几何的测地线。因此，我们的正则化将轨迹的欧几里得几何替换为适当的信息几何。通过将欧拉方程视为微分同胚流形上的动力系统，我们将这一思想推广到无限族路径。我们的正则化将该流形嵌入信息几何环境空间，为其配备测地完备的几何结构。将得到的拉格朗日方程表示为欧拉形式，我们推导出守恒形式的正则化欧拉方程。在一维和二维问题上的数值实验展示了其作为数值工具的潜力。尽管我们为了具体性和表述简洁性而聚焦于正压欧拉方程，但我们的正则化方法很容易推广到更一般的欧拉型和纳维-斯托克斯型方程。