We are concerned with the numerical solution of a linear transport problem with nonzero divergence velocity field that originates from the spectral energy balance equation describing the evolution of wind waves and swells in coastal seas. The discretization error of the commonly used first‐order upwind finite difference and first‐order vertex‐centered upwind finite volume schemes in one space dimension is analyzed. Smoothness of nondivergent velocity field plays a crucial role in this. No such analysis has been attempted to date for such problems. The two schemes studied differ in the manner in which they treat the scalar flux numerically. The finite difference variant is shock captured, whereas the vertex‐centered finite volume approximation employs an arithmetic mean of the velocity and appears not to be flux conservative. The methods are subsequently extended to two dimensions on triangular meshes. Numerical experiments are provided to verify the convergence analysis. The main finding is that the finite difference scheme displays optimal rates of convergence and offers higher accuracy over the finite volume scheme, regardless the regularity of the velocity field. The latter scheme notably yields convergence rates of 0.5 and 0 in L²‐norm and L^∞‐norm, respectively, when the velocity field is not smooth. A test case illustrating wave shoaling and refraction over submerged shoals is also presented and demonstrates the practical importance of flux conservation.

中文翻译：

---

由能量平衡方程引起的具有零散度速度场的标量输运的传统离散化方法的精度方面

我们关注具有非零发散速度场的线性输运问题的数值解，该解源自于描述沿海海域风波和涌浪演变的频谱能量平衡方程。分析了在一个空间维度上常用的一阶迎风有限差分和一阶顶点为中心的迎风有限体积方案的离散误差。非发散速度场的平滑度在其中起着至关重要的作用。迄今为止，尚未尝试针对此类问题进行此类分析。所研究的两种方案在数值上处理标量通量的方式不同。有限差分变量是捕捉到的震动，而以顶点为中心的有限体积近似采用速度的算术平均值，并且似乎不是磁通保守的。该方法随后在三角形网格上扩展到二维。提供数值实验以验证收敛性分析。主要发现是，无论速度场的规则性如何，有限差分方案都显示出最佳的收敛速度，并且比有限体积方案提供更高的精度。后一种方案的收敛速度显着为0.5和0。大号²范数和大号^∞范数，分别，当速度场不光滑。还提供了一个测试案例，说明了波浪在浅滩上的浅滩和折射现象，并说明了通量守恒的实际重要性。