Maximum-principle-satisfying second-order Intrusive Polynomial Moment scheme
This work addresses the need for efficient and stable numerical schemes for uncertainty quantification in hyperbolic conservation laws, offering a second-order accurate method that maintains physical bounds.
The authors derive a second-order discretization of the Intrusive Polynomial Moment (IPM) method for hyperbolic conservation laws with uncertainties that satisfies the maximum principle, reducing numerical costs while ensuring nonoscillatory solutions.
Using standard intrusive techniques when solving hyperbolic conservation laws with uncertainties can lead to oscillatory solutions as well as nonhyperbolic moment systems. The Intrusive Polynomial Moment (IPM) method ensures hyperbolicity of the moment system while restricting oscillatory over- and undershoots of specified bounds. In this contribution, we derive a second-order discretization of the IPM moment system which fulfills the maximum principle. This task is carried out by investigating violations of the specified bounds due to the errors from the numerical optimization required by the scheme. This analysis gives weaker conditions on the entropy that is used, allowing the choice of an entropy which enables choosing the exact minimal and maximal value of the initial condition as bounds. Solutions calculated with the derived scheme are nonoscillatory while fulfilling the maximum principle. The second-order accuracy of our scheme leads to significantly reduced numerical costs.