We formulate an oversampled radial basis function generated finite difference (RBF-FD) method to solve time-dependent nonlinear conservation laws. The analytic solutions of these problems are known to be discontinuous, which leads to occurrence of non-physical oscillations (Gibbs phenomenon) that pollute the numerical solutions and can make them unstable. We address these difficulties using a residual based artificial viscosity stabilization, where the residual of the conservation law indicates the approximate location of the shocks. The location is then used to locally apply an upwind viscosity term, which stabilizes the Gibbs phenomenon and does not smear the solution away from the shocks. The proposed method is numerically tested and proves to be robust and accurate when solving scalar conservation laws and systems of conservation laws, such as compressible Euler equations.

中文翻译：

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用于求解非线性守恒定律的残余粘度稳定 RBF-FD 方法

我们制定了一种过采样径向基函数生成有限差分 (RBF-FD) 方法来求解与时间相关的非线性守恒定律。已知这些问题的解析解是不连续的，这会导致非物理振荡（吉布斯现象）的发生，这些振荡会污染数值解并使其不稳定。我们使用基于残差的人工粘度稳定来解决这些困难，其中守恒定律的残差表示冲击的大致位置。然后使用该位置局部应用逆风粘度项，这可以稳定吉布斯现象，并且不会将解决方案从冲击中抹去。所提出的方法经过数值测试，证明在求解标量守恒定律和守恒定律系统时是稳健和准确的，