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 an inviscid regularization based on ideas from semidefinite programming and information geometry. 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 nonintersecting 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 amounts to replacing the Euclidean geometry of phase space 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.

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”解通常通过黏性正则化来定义，但即使微小的黏性也可能显著改变解的长期行为。在本工作中，我们提出了一种基于半定规划和信息几何思想的无黏正则化方法。从拉格朗日视角来看，熵解中的激波形成相当于流体粒子的非弹性碰撞。其轨迹类似于在非交叉路径可行集上的投影梯度下降路径。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法解路径来进行正则化。这些路径是由障碍函数诱导的信息几何下的测地线。因此，我们的正则化实质上是将相空间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学系统，我们将这一思想推广到无穷多路径族。该正则化将该流形嵌入到信息几何的赋空间，使其具有测地完备的几何。将所得的拉格朗日方程表达为欧拉形式，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。