An approximate Jacobian nonlinear solver for multiphase flow and transport
An approximate Jacobian nonlinear solver for multiphase flow and transport
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用于多相流和输运的近似雅可比非线性求解器
DOI：
10.1016/j.jcp.2018.08.043
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
M. Wheeler
中科院分区：
文献类型：
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作者：
Gurpreet Singh;G. Pencheva;M. Wheeler
我们提出了一种近似雅可比法来求解多孔介质中非线性多相流动和输运问题。在采用最低阶混合有限元法（MFEM）进行空间离散之前，先采用倒推欧拉时间离散格式。这就得到了一个完全隐式的非线性代数方程组。在牛顿线性化过程中，通常采用精确的雅可比矩阵构造来获得经过时空离散化的线性方程组。这种完全耦合的单片线性系统，通常在压力和饱和度（或浓度）未知的情况下，需要专门的预调节器，如约束压力残余（CPR）或两级预调节器。这些预调节器在线性系统上运行，以解耦压力和饱和（或浓度）自由度（DOF），以便使用现有的线性求解器来求解正定（PD）矩阵，如GMRES或AMG，仅举几例。在这项工作中，我们提出了一种替代两阶段预处理（或CPR）的方法来解决牛顿线性化后的上述单片系统。该方法依赖于牛顿线性化过程中压力饱和度（或浓度）块子矩阵的解耦近似，以获得块对角子矩阵。由此产生的线性系统很容易减少，简单地消除这些对角子矩阵，以获得一个系统的压力自由度，避免需要专门的预调节器。此外，由于消除了饱和（或浓度）未知数，线性系统具有较小的自由度。该非线性解算器被证明与精确的雅可比方法一样精确，根据两种方法的非线性残差收敛到所需的公差来测量。我们的数值结果表明，对于所考虑的两相流模型公式，计算速度提高了约1.32至4.0倍。这与近似雅可比法和精确雅可比法下线性系统的自由度有关。对于多组分流和输运，这种加速预期与浓度自由度的数量成正比。若干场尺度的数值模拟也证明了该方法对实际问题的有效性。
We present an approximate Jacobian approach for solving nonlinear, multiphase flow and transport problems in porous media. A backward Euler time discretization scheme is used prior to spatial discretization with a lowest order mixed finite element method (MFEM). This results in a fully implicit nonlinear algebraic system of equations. Conventionally, an exact Jacobian construction is employed during the Newton linearization to obtain a linear system of equations after spatial and temporal discretization. This fully coupled, monolithic linear system, usually in pressure and saturation (or concentration) unknowns, requires specialized preconditioners such as constrained pressure residual (CPR) or two stage preconditioner. These preconditioners operate on the linear system to decouple pressure and saturation (or concentration) degrees of freedom (DOF) in order to use existing linear solvers for positive definite (PD) matrices such as GMRES or AMG, to name a few. In this work, we present an alternative to two-stage preconditioning (or CPR) for solving the aforementioned monolithic system after Newton linearization. This approach relies upon a decoupling approximation for the pressure-saturation (or concentration) block sub-matrices, during Newton linearization, to obtain block diagonal sub-matrices. The resulting linear system is easily reduced, trivially eliminating these diagonal sub-matrices, to obtain a system in pressure DOF circumventing the need for specialized preconditioners. Further, the linear system has lesser DOF owing to the elimination of saturation (or concentration) unknowns. This nonlinear solver is demonstrated to be as accurate as the exact Jacobian approach, measured in terms of convergence of nonlinear residual to a desired tolerance for both methods. Our numerical results indicate a consistent computational speedup by a factor of approximately 1.32 to 4.0 for the two-phase flow model formulation under consideration. This is related to the DOF of the linear systems for the approximate and exact Jacobian approaches. For multicomponent flow and transport, this speedup is expected to be directly proportional to the number of concentration degrees of freedom. A number of field scale numerical simulations are also presented to demonstrate the efficacy of this approach for realistic problems.
日本儿童和学生对 TIMSS 科学论文式任务的反应特征 (7) - 9 项任务分析结果的趋势 -
DOI：
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发表时间：
2005
期刊：
日本科学教育学会年会論文集 第29号
影响因子：
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作者：
中山 迅;大場裕子;猿田祐嗣
通讯作者：
猿田祐嗣