A fourth-order approximate projection method for the incompressible Navier–Stokes equations on locally-refined periodic domains

In this follow-up of our previous work [30], the author proposes a high-order semi-implicit method for numerically solving the incompressible Navier–Stokes equations on locally-refined periodic domains. Fourth-order finite-volume stencils are employed for spatially discretizing various operators in the context of structured adaptive mesh refinement (AMR). Time integration adopts a fourth-order, semi-implicit, additive Runge–Kutta method to treat the non-stiff convection term explicitly and the stiff diffusion term implicitly. The divergence-free condition is fulfilled by an approximate projection operator. Altogether, these components yield a simple algorithm for simulating incompressible viscous flows on periodic domains with fourth-order accuracies both in time and in space. Results of numerical tests show that the proposed method is superior to previous second-order methods in terms of accuracy and efficiency. A major contribution of this work is the analysis of a fourth-order approximate projection operator.     

一维有限元的EEP单元及其自适应分析

对m(>1)次单元,基于单元能量投影(element energy projection,简称EEP)法提出的简约格式位移解u^*具有比常规有限元解u^h至少高一阶的精度,据此提出了EEP单元概念,并给出以EEP单元作为最终解的自适应有限元求解策略. 通过编制相应的计算程序分析了一维非自伴随问题,计算结果与理论预期吻合较好,验证了自适应求解策略的有效性和可靠性. 研究结果表明：该法可以给出按最大模度量、逐点满足误差限的解答,相较于常规单元,最终的求解单元数更少.