Rémi Abgrall, Fatemeh Nassajian Mojarrad
We are interested in building schemes for the compressible Euler equations that are also locally conserving the angular momentum. We present a general framework, describe a few examples of schemes and show results. These schemes can be of arbitrary order.
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Conservation properties of iterative methods applied to implicit finite volume discretizations of nonlinear conservation laws are analyzed. It is shown that any consistent multistep or Runge-Kutta method is globally conservative. Further, it is shown that Newton's method, Krylov subspace methods and pseudo-time iterations are globally conservative while the Jacobi and Gauss-Seidel methods are not in general. If pseudo-time iterations using an explicit Runge-Kutta method are applied to a locally conservative discretization, then the resulting scheme is also locally conservative. However, the co…
Flux-conservative Hermite methods for simulation of nonlinear conservation laws
本文提出了一类新的赫尔米特方法，保留高阶空间精度的同时提升稳定性，通过熵粘性方法捕捉激波。
Locally conservative redistribution limiting and applications to the approximation of conservation equations, Part II
提出一种非侵入式、与离散化和PDE无关的限制技术，具有守恒性，适用于高阶近似（包括谱方法），可用于时间依赖和独立问题以及椭圆、抛物和双曲问题。
A Globally Conservative Compact Framework for Conservation Laws: Fourth-Order Schemes with Enhanced Resolution and Stability
本文提出一种全局保守紧致差分方案框架，通过多项式空间中的最优代数阶实现严格守恒，提升四阶方案的分辨率和稳定性。
Reduced order models of nonlinear conservation laws in fluid dynamics do not typically inherit stability properties of the full order model. We introduce projection-based hyper-reduced models of nonlinear conservation laws which are globally conservative and inherit a semi-discrete entropy inequality independently of the choice of basis and choice of parameters.
In this paper, we solve nonlinear conservation laws using the radial basis function generated finite difference (RBF-FD) method. Nonlinear conservation laws have solutions that entail strong discontinuities and shocks, which give rise to numerical instabilities when the solution is approximated by a numerical method. We introduce a residual-based artificial viscosity (RV) stabilization framework adjusted to the RBF-FD method, where the residual of the conservation law adaptively locates discontinuities and shocks. The RV stabilization framework is applied to the collocation RBF-FD method and t…
Bespoke finite difference methods that preserve two local conservation laws of the modified KdV equation
本文提出了一种新的有限差分方法，通过利用守恒律作为离散欧拉算子核的性质，构造出能保留质量与能量局部守恒律的定制方法。
This paper introduces a new symbolic-numeric strategy for finding semidiscretizations of a given PDE that preserve multiple local conservation laws. We prove that for one spatial dimension, various one-step time integrators from the literature preserve fully discrete local conservation laws whose densities are either quadratic or a Hamiltonian. The approach generalizes to time integrators with more steps and conservation laws of other kinds; higher-dimensional PDEs can be treated by iterating the new strategy. We use the Boussinesq equation as a benchmark and introduce new families of schemes …
Hypernetwork-Conditioned WENO5 Conservative-Form CNNs for One-Dimensional Conservation Laws
本文提出一种基于经典WENO有限体积方案和超网络架构的保守数据驱动离散方法，通过轻量目标网络预测非线性WENO权重，实现适应不同问题实例和空间分辨率的高阶学习。
In this paper we blend high-order Compact Approximate Taylor (CAT) numerical methods with the a posteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm to solve hyperbolic systems of conservation laws. The resulting methods are highly accurate for smooth solutions, essentially non-oscillatory for discontinuous ones, and almost fail-safe positivity preserving. Some numerical results for scalar conservation laws and systems are presented to show the appropriate behavior of CAT-MOOD methods.
The paper considers one-dimensional flows of a polytropic gas in the Lagrangian coordinates in three cases: plain one-dimensional flows, radially symmetric flows and spherically symmetric flows. The one-dimensional flow of a polytropic gas is described by one second-order partial differential equation in the Lagrangian variables. Lie group classification of this PDE is performed. Its variational structure allows to construct conservation laws with the help of Noether's theorem. These conservation laws are also recalculated for the gas dynamics variables in the Lagrangian and Eulerian coordinat…
Solving 1D Conservation Laws Using Pontryagin's Minimum Principle
本文探讨了标量凸守恒律与庞特里亚金最优化原理之间的联系，提出了一种求解守恒律的最小值解方法，并证明该解等价于熵解，扩展了Lax-Oleinik公式至非均匀凸通量函数。
A class of high-order discontinuous-Galerkin methods satisfying infinitely many entropy conditions with provable error estimates and strong convergence for general nonlinear conservation laws
本文提出基于算子半群的半离散DG方法新框架，构造满足无穷多局部熵不等式的高阶OFDG型格式，推广至任意维度守恒律系统，证明了光滑解最优误差估计与严格凸守恒律间断解的强收敛性。
An exactly conservative particle method for one dimensional scalar conservation laws
提出一种基于特征速度移动粒子、局部解析相互作用实现精确守恒的粒子方法，用于一维标量守恒律，具有变差减小、熵减、无数值耗散和二阶精度等特性。
We study a class of variational problems for regularized conservation laws with Lax's entropy-entropy flux pairs. We first introduce a modified optimal transport space based on conservation laws with diffusion. Using this space, we demonstrate that conservation laws with diffusion are "flux--gradient flows". We next construct variational problems for these flows, for which we derive dual PDE systems for regularized conservation laws. Several examples, including traffic flow and Burgers' equation, are presented. Incorporating both primal-dual algorithms and monotone schemes, we successfully com…
Neural Entropy-stable conservative flux form neural networks for learning hyperbolic conservation laws
本文提出神经熵稳定守恒通量形式神经网络（NESCFN），通过联合学习数值通量和熵函数，在完全数据驱动下学习双曲守恒律，实现长期稳定守恒并准确捕捉激波速度。
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