An adaptive local time-stepping scheme for multiresolution simulations of hyperbolic conservation laws
An adaptive local time-stepping scheme for multiresolution simulations of hyperbolic conservation laws
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DOI：
10.1016/j.jcpx.2019.100038
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发表时间：
2019-09
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影响因子：
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通讯作者：
J. Kaiser;N. Hoppe;S. Adami;N. Adams
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作者：
J. Kaiser;N. Hoppe;S. Adami;N. Adams
提出了一种求解流体双曲守恒律方程块结构多分辨格式的自适应局部时间步（ALTS）格式。标准的本地时间步长（LTS）计划与水平相关的时间步长的稳定性提高了本地时间步长的适应时，通过底层的多阶段时间积分计划。该方法的新奇在于，它将通量计算和状态向量的时间积分与多分辨率方案的投影和预测操作合并在一起[15]。这使得具有不同细化级别的子域的一致时间积分成为可能，而不需要在并行计算中可能过于昂贵的中间时间同步。因此，只有当较细的子域前进到同一时刻时，较粗的子域才会在时间上前进。新格式在每个子步后都保持了积分区域的全空间分辨率自适应性，并且由于每个子步的局部时间步长自适应，与以前的LTS格式相比，新格式的数值稳定性有了显著的提高。所引起的额外操作的计算开销很小。在实际应用中，ALTS格式具有与标准LTS格式相同的计算效率，并且可以应用于任何显式的单步时间积分格式，与所采用的空间离散格式无关。改进的稳定性证明了几个一维和二维的例子与一个和两个阶段的流动，应用二阶和三阶龙格-库塔时间积分方案。
We present an adaptive local time-stepping (ALTS) scheme for a block-structured multiresolution scheme of hyperbolic conservation laws for fluid flow. The stability of standard local time-stepping (LTS) schemes with level-dependent time-step sizes is improved by local time-step size adaptation when progressing through the underlying multi-stage time integration scheme. The novelty of the approach is that it merges flux computation and time integration of the state vector with projection and prediction operations of the multiresolution scheme [15]. This enables consistent time integration of subdomains with different refinement levels without the need for intermediate time synchronization which can be prohibitively expensive in parallel computations. Consequently, coarser subdomains are advanced in time only once finer subdomains have advanced to the same time instant. Full spatial resolution adaptivity for integrated regions after each substep is maintained.The new scheme exhibits significantly improved numerical stability as compared to previous LTS schemes due to the local time-step size adaptation at each substep. The computational overhead of the incurred additional operations is small. In applications, the ALTS scheme demonstrates the same computational efficiency as standard LTS schemes.The new scheme can be applied to any explicit single-step time-integration scheme and is independent of the employed spatial discretization scheme. The improved stability is demonstrated with several one- and two-dimensional examples of flows with one and two phases, applying second- and third-order Runge-Kutta time integration schemes.