High-Resolution Multimesh hp-FEM for Simulation of Compressible Particle-Laden Gas Flows

该项目的目标是在第一个资助期间实施的方法的基础上，发展一种新的方法来研究标量输运方程和双曲系统的高温自适应性。所提出的变分多尺度框架将粗尺度的连续伽辽金近似与细尺度的不连续近似结合起来。为此，连续（P1或Q1）有限元将被用于设计p自适应不连续Galerkin （DG）方法的分层斜率限制技术的Taylor基函数所丰富。在粗尺度上，离散最大原则将使用FCT型（通量校正输运）的代数通量校正方案来强制执行。在HERMES软件包中实现的多网格hp-FEM方法使得可以为每个变量和p-层次的每个级别使用单独设计的网格。利用最大原则估计局部正则性，确定自适应类型（h或p）。平滑和误差估计的过程涉及到对最高阶导数的重建技术的使用。将非线性系统分解为粗尺度的全局问题和细尺度的小局部问题，可以大大提高效率。将新的hp-FEM扩展到欧拉方程和双流体模型方程将建立在先前开发的用于粗尺度模拟的数值算法的基础上。

The objective of this project is the development of a new approach to hp-adaptivity for scalar transport equations and hyperbolic systems on the basis of methods implemented during the first funding period. The proposed variational multiscale framework will combine a continuous Galerkin approximation of coarse scales with a discontinuous approximation of fine scales. To this end, continuous (P1 or Q1) finite elements will be enriched by Taylor basis functions that have been employed to design hierarchichal slope limiting techniques for p-adaptive discontinuous Galerkin (DG) methods. At the coarse scale level, the discrete maximum principle will be enforced using an algebraic flux correction scheme of FCT type (Flux-Corrected Transport). The multimesh hp-FEM approach implemented in the software package HERMES makes it possible to use individually designed meshes for each variable and each level of the p-hierarchy. The type of adaptivity (h or p) is determined using maximum principles to estimate the local regularity. The process of smoothness and error estimation involves the use of reconstruction techniques for the highest-order derivatives. The decomposition of the nonlinear system into a global problem for the coarse scales and small local problems for the fine scales offers substantial efficiency gains. The extension of the new hp-FEM to the Euler equations and equations of the two-fluid modell will build on the previously developed numerical algorithms for coarse-scale simulations.