In [Cou15] a multiplier technique, going back to Leray and G{\aa}rding for scalar hyperbolic partial differential equations, has been extended to the context of finite difference schemes for evolutionary problems. The key point of the analysis in [Cou15] was to obtain a discrete energy-dissipation balance law when the initial difference operator is multiplied by a suitable quantity. The construction of the energy and dissipation functionals was achieved in [Cou15] under the assumption that all modes were separated. We relax this assumption here and construct, for the same multiplier as in [Cou15], the energy and dissipation functionals when some modes cross. Semigroup estimates for fully discrete hy-perbolic initial boundary value problems are deduced in this broader context by following the arguments of [Cou15].

中文翻译：

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有限差分格式的 Leray-G{\aa}rding 方法。二、平滑交叉模式

在 [Cou15] 中，一种乘法器技术，可以追溯到标量双曲偏微分方程的 Leray 和 G{\aaa}rding，已扩展到进化问题的有限差分格式的上下文中。[Cou15]中分析的重点是在初始差分算子乘以合适的量时获得离散的耗能平衡律。能量和耗散函数的构建是在 [Cou15] 中在所有模式分离的假设下实现的。我们在这里放宽了这个假设，并为与 [Cou15] 中相同的乘数构造了一些模式交叉时的能量和耗散函数。通过遵循 [Cou15] 的论点，在更广泛的背景下推导出完全离散双曲初始边界值问题的半群估计。