Abstract In this article we present a novel space-time semi-Lagrangian advection scheme for the solution of the nonlinear convective terms in hyperbolic conservation laws. The governing equations are discretized on a three-dimensional mesh, composed of a staggered unstructured Voronoi grid on the horizontal plane which is extruded along the vertical direction with z − layers of non-uniform thickness. A high order space-time reconstruction is carried out for the velocity field, that is used for both tracking backward in time the Lagrangian trajectories of the flow and for the interpolation of the transported quantity at the foot of the characteristics. High order in space is achieved via a constrained least-squares reconstruction technique, whereas the ADER procedure is employed for gaining high order of accuracy in time as well. The high order reconstruction polynomials are expanded onto a set of basis functions that are defined in the physical coordinate system for space and in the reference framework for time, thus improving the computational efficiency of the scheme. The trajectory equation of the flow particles is then solved relying on symplectic-type integrators, which are proven to be structure-preserving ODE solvers, unlike standard explicit Runge–Kutta schemes. Application to hydrostatic free surface flows is proposed, demonstrating accuracy and robustness of the novel numerical method via comparison against analytical solutions.

中文翻译：

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应用于自由表面流的交错 Voronoi 网格的时空半拉格朗日平流方案

摘要 在本文中，我们提出了一种新的时空半拉格朗日平流方案，用于求解双曲守恒定律中的非线性对流项。控制方程在三维网格上离散化，由水平面上的交错非结构化 Voronoi 网格组成，该网格沿垂直方向挤压，具有非均匀厚度的 z - 层。对速度场进行了高阶时空重建，用于在时间上向后跟踪流动的拉格朗日轨迹和在特征的底部对输送量进行插值。空间上的高阶是通过约束最小二乘重建技术实现的，而 ADER 过程也用于在时间上获得高阶精度。将高阶重构多项式扩展到一组在空间物理坐标系和时间参考框架中定义的基函数，从而提高了方案的计算效率。然后依靠辛型积分器求解流动粒子的轨迹方程，与标准显式 Runge-Kutta 方案不同，辛型积分器被证明是结构保持型 ODE 求解器。提出了对静水自由表面流动的应用，通过与解析解的比较证明了新型数值方法的准确性和鲁棒性。然后依靠辛型积分器求解流动粒子的轨迹方程，与标准显式 Runge-Kutta 方案不同，辛型积分器被证明是结构保持型 ODE 求解器。提出了对静水自由表面流动的应用，通过与解析解的比较证明了新型数值方法的准确性和鲁棒性。然后依靠辛型积分器求解流动粒子的轨迹方程，与标准显式 Runge-Kutta 方案不同，辛型积分器被证明是结构保持型 ODE 求解器。提出了对静水自由表面流动的应用，通过与解析解的比较证明了新型数值方法的准确性和鲁棒性。