We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton's method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers' equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.

翻译：我们考虑非线性守恒律的隐式时间离散格式中熵守恒与耗散离散化，并研究求解非线性方程时所用迭代方法的影响。研究表明，即使迭代误差小于时间积分误差，牛顿法仍可能将熵耗散格式转变为反耗散格式。我们探索了几种补救措施，其中性能最优的是松弛技术——该技术最初设计用于修正时间积分方法中的熵误差。因此，只要迭代误差达到时间积分方法的量级，松弛技术便能与迭代求解器有效配合。为验证这些发现，我们考虑了Burgers方程与非线性色散波动方程。结果表明，即使容差大一个数量级，熵守恒格式相比非守恒格式仍能获得更精确的数值解。