We consider nonlinear scalar conservation laws posed on a network. We establish $L^1$ stability, and thus uniqueness, for weak solutions satisfying the entropy condition. We apply standard finite volume methods and show stability and convergence to the unique entropy solution, thus establishing existence of a solution in the process. Both our existence and stability/uniqueness theory is centred around families of stationary states for the equation. In one important case -- for monotone fluxes with an upwind difference scheme -- we show that the set of (discrete) stationary solutions is indeed sufficiently large to suit our general theory. We demonstrate the method's properties through several numerical experiments.
翻译:我们考虑的是网络上的非线性天平保护法。我们为满足酶质条件的薄弱解决方案建立1美元的稳定,因此是独一无二的。我们采用标准的有限体积方法,并表现出稳定性和趋同性,从而在这个过程中确立了一种解决办法。我们的存在和稳定性/独特性理论都集中在固定状态状态的等式家庭。在一个重要的例子中,对于单体通量和上风差异方案 -- -- 我们表明,(分辨性)固定式解决方案的组装确实足够大,足以符合我们的一般理论。我们通过几个数字实验来证明该方法的特性。