弯曲时空上的相对论Burgers方程：推导与有限体积逼近
Relativistic Burgers equations on curved spacetimes. Derivation and finite volume approximation
AI总结：
本文在弯曲时空中识别出一类具有洛伦兹不变性的非线性双曲平衡律，推导了相对论Burgers方程，并提出了保持稳态解的几何有限体积格式。
摘要：
在弯曲时空（配备体积形式）上的一类非线性双曲平衡律中，我们识别出一个具有与相对论可压缩流体欧拉方程相同的洛伦兹不变性的双曲平衡律。该模型在归一化意义下唯一，并在光速无穷大的极限下收敛到标准无粘Burgers方程。此外，从弯曲背景上的相对论可压缩流欧拉系统，我们推导了标准无粘Burgers方程及其相对论推广。所提出的模型被称为弯曲时空上的相对论Burgers方程，为我们提供了简单模型，便于数值方法的开发和分析。接下来，我们引入一种有限体积格式来逼近这些相对论Burgers方程的不连续解。我们的格式在几何上构造，并与所考虑平衡律的自然散度形式一致。它适用于包含激波的弱解，并且最重要的是，它是良好平衡的，即保持稳态解。数值实验展示了所提有限体积格式的收敛性及其在计算弯曲背景上熵解的相关性。
英文摘要：
Within the class of nonlinear hyperbolic balance laws posed on a curved spacetime (endowed with a volume form), we identify a hyperbolic balance law that enjoys the same Lorentz invariance property as the one satisfied by the Euler equations of relativistic compressible fluids. This model is unique up to normalization and converges to the standard inviscid Burgers equation in the limit of infinite light speed. Furthermore, from the Euler system of relativistic compressible flows on a curved background, we derive, both, the standard inviscid Burgers equation and our relativistic generalizations. The proposed models are referred to as relativistic Burgers equations on curved spacetimes and provide us with simple models on which numerical methods can be developed and analyzed. Next, we introduce a finite volume scheme for the approximation of discontinuous solutions to these relativistic Burgers equations. Our scheme is formulated geometrically and is consistent with the natural divergence form of the balance laws under consideration. It applies to weak solutions containing shock waves and, most importantly, is well-balanced in the sense that it preserves steady solutions. Numerical experiments are presented which demonstrate the convergence of the proposed finite volume scheme and its relevance for computing entropy solutions on a curved background.
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