We introduce stabilized spline collocation schemes for the numerical solution of nonlinear, hyperbolic conservation laws. A nonlinear, residual-based viscosity stabilization is combined with a projection stabilization-inspired linear operator to stabilize the scheme in the presence of shocks and prevent the propagation of spurious, small-scale oscillations. Due to the nature of collocation schemes, these methods possess the possibility for greatly reduced computational cost of high-order discretizations. Numerical results for the linear advection, Burgers, Buckley-Leverett, and Euler equations show that the scheme is robust in the presence of shocks while maintaining high-order accuracy on smooth problems.

翻译：本文提出了稳定化样条配点格式，用于数值求解非线性双曲守恒律。将基于残差的非线性黏性稳定化与受投影稳定化启发的线性算子相结合，以在激波存在时稳定格式并防止虚假的小尺度振荡传播。由于配点格式的特性，这些方法具有大幅降低高阶离散计算成本的潜力。针对线性对流方程、Burgers方程、Buckley-Leverett方程和Euler方程的数值结果表明，该格式在激波存在时具有鲁棒性，同时在光滑问题上保持高阶精度。