For hyperbolic conservation laws, the famous Lax-Wendroff theorem delivers sufficient conditions for the limit of a convergent numerical method to be a weak (entropy) solution. This theorem is a fundamental result, and many investigations have been done to verify its validity for finite difference, finite volume, and finite element schemes, using either explicit or implicit linear time-integration methods. Recently, the use of modified Patankar (MP) schemes as time-integration methods for the discretization of hyperbolic conservation laws has gained increasing interest. These schemes are unconditionally conservative and positivity-preserving and only require the solution of a linear system. However, MP schemes are by construction nonlinear, which is why the theoretical investigation of these schemes is more involved. We prove an extension of the Lax-Wendroff theorem for the class of MP methods. This is the first extension of the Lax--Wendroff theorem to nonlinear time integration methods with just an additional hypothesis on the total time variation boundedness of the numerical solutions. We provide some numerical simulations that validate the theoretical observations.

翻译：对于双曲守恒律，著名的Lax-Wendroff定理为收敛数值方法的极限成为弱（熵）解提供了充分条件。该定理是基础性结果，已有大量研究验证其在有限差分、有限体积和有限元格式中的有效性，这些格式采用显式或隐式线性时间积分方法。近年来，采用修正Patankar（MP）格式作为双曲守恒律离散化的时间积分方法日益受到关注。这些格式具有无条件守恒性和保正性，且仅需求解线性系统。然而，MP格式在构造上具有非线性特征，导致其理论分析更为复杂。我们证明了MP方法类别的Lax-Wendroff定理扩展形式，这是该定理首次推广至非线性时间积分方法，仅需额外假设数值解的总时间变差有界性。我们通过数值模拟验证了理论观测结果。