Abstract In this paper, we propose a novel trajectory-approximation technique as a time-integration scheme in a semi-Lagrangian framework, which is generally applicable to solve advectional partial differential equations in engineering and physics. The proposed trajectory-approximation technique resolves strong nonlinearity in the Cauchy problem and saves computational costs in comparison with the existing third-order methods by reducing the number of interpolations occurring at every spatial lattice point for each time step. Moreover, an explicit formula is introduced as a more efficient form of the proposed time-integration scheme. To obtain numerical evidence, we apply the proposed method to simulate four benchmark test flows of incompressible Navier–Stokes equations: a linear advection–diffusion, a flow on a square domain, a shear layer flow, and a backward-facing step flow. The proposed method provides third-order accuracy in terms of both time and space in the overall backward semi-Lagrangian methodology. It also demonstrates superior performance over recently developed third-order trajectory-approximation schemes in terms of the efficiency and execution time in solving the Cauchy problem with strong nonlinearity.

中文翻译：

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用于求解不可压缩 Navier-Stokes 方程的降低成本的时间积分方案的算法

摘要 在本文中，我们提出了一种新的轨迹逼近技术作为半拉格朗日框架中的时间积分方案，它通常适用于求解工程和物理学中的对流偏微分方程。所提出的轨迹近似技术解决了柯西问题中的强非线性问题，与现有的三阶方法相比，通过减少每个时间步长在每个空间格点上发生的插值次数，节省了计算成本。此外，引入了显式公式作为所提出的时间积分方案的更有效形式。为了获得数值证据，我们应用所提出的方法来模拟不可压缩 Navier-Stokes 方程的四种基准测试流：线性对流-扩散、方形域上的流、剪切层流、和一个向后的步骤流程。所提出的方法在整体后向半拉格朗日方法中在时间和空间方面都提供了三阶精度。在解决具有强非线性的柯西问题的效率和执行时间方面，它还表现出优于最近开发的三阶轨迹近似方案的性能。