We consider a one-dimensional nonlocal hyperbolic model introduced to describe the formation and movement of self-organizing collectives of animals in homogeneous 1D environments. Previous research has shown that this model exhibits a large number of complex spatial and spatiotemporal aggregation patterns, as evidenced by numerical simulations and weakly nonlinear analysis. In this study, we focus on a particular type of localised patterns with odd/even/no symmetries (which are usually part of snaking solution branches with different symmetries that form complex bifurcation structures called snake-and-ladder bifurcations). To numerically investigate the bifurcating solution branches (to eventually construct the full bifurcating structures), we first need to understand the numerical issues that could appear when using different numerical schemes. To this end, in this study, we consider ten different numerical schemes (the upwind scheme, the MacCormack scheme, the Fractional-Step method, and the Quasi-Steady Wave-Propagation algorithm, combining them with high-resolution methods), while paying attention to the preservation of the solution symmetries with all these schemes. We show several numerical issues: first, we observe the presence of two distinct types of numerical solutions (with different symmetries) that exhibit very small errors; second, in some cases, none of the investigated numerical schemes converge, posing a challenge for the development of numerical continuation algorithms for nonlocal hyperbolic systems; lastly, the choice of the numerical schemes, as well as their corresponding parameters such as time-space steps, exert a significant influence on the type and symmetry of bifurcating solutions.

翻译：我们考虑一维非局部双曲模型，该模型旨在描述同质一维环境中动物自组织团体的形成与运动。先前研究表明，通过数值模拟和弱非线性分析，该模型展现出大量复杂的空间及时空聚集模式。本研究聚焦于具有奇对称/偶对称/无对称性的特定类型局域模式（这些模式通常属于不同对称性的蜿蜒解分支，形成称为梯蛇分岔的复杂分岔结构）。为数值研究分岔解分支（最终构建完整分岔结构），我们首先需要理解使用不同数值方案时可能出现的数值问题。为此，本研究考察了十种数值方案（迎风格式、MacCormack格式、分数步法、准稳态波传播算法，并结合高分辨率方法），同时关注所有方案对解对称性的保持情况。我们揭示了若干数值问题：首先，观测到存在两类具有极小误差的数值解（具不同对称性）；其次，在某些情况下，所研究的数值方案均不收敛，这对非局部双曲系统数值延拓算法的开发构成挑战；最后，数值方案的选择及其对应参数（如时空步长）显著影响分岔解的类型与对称性。