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：我们研究了隐式时间离散格式下非线性守恒律的熵守恒与熵耗散离散化，并探讨了求解所产生非线性方程时使用的迭代方法的影响。我们证明，即使迭代误差小于时间积分误差，牛顿法也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——该技术最初为修正时间积分方法中的熵误差而设计。因此，只要迭代误差与时间积分方法的误差量级相当，松弛技术便能与迭代求解器良好配合。为验证结论，我们考虑了Burgers方程与非线性色散波动方程。我们发现，即使容差大一个数量级，熵守恒格式仍比非守恒格式能获得更精确的数值解。