In this work, we use the monolithic convex limiting (MCL) methodology to enforce relevant inequality constraints in implicit finite element discretizations of the compressible Euler equations. In this context, preservation of invariant domains follows from positivity preservation for intermediate states of the density and internal energy. To avoid spurious oscillations, we additionally impose local maximum principles on intermediate states of the density, velocity components, and specific total energy. For the backward Euler time stepping, we show the invariant domain preserving (IDP) property of the fully discrete MCL scheme by constructing a fixed-point iteration that is IDP and converges under a strong time step restriction. Our iterative solver for the nonlinear discrete problem employs a more efficient fixed-point iteration. The matrix of the associated linear system is a robust low-order Jacobian approximation that exploits the homogeneity property of the flux function. The limited antidiffusive terms are treated explicitly. We use positivity preservation as a stopping criterion for nonlinear iterations. The first iteration yields the solution of a linearized semi-implicit problem. This solution possesses the discrete conservation property but is generally not IDP. Further iterations are performed if any non-IDP states are detected. The existence of an IDP limit is guaranteed by our analysis. To facilitate convergence to steady-state solutions, we perform adaptive explicit underrelaxation at the end of each time step. The calculation of appropriate relaxation factors is based on an approximate minimization of nodal entropy residuals. The performance of proposed algorithms and alternative solution strategies is illustrated by the convergence history for standard two-dimensional test problems.

翻译：本文采用整体凸限制（MCL）方法，在可压缩欧拉方程的隐式有限元离散格式中实施相关不等式约束。在此框架下，密度与内能中间状态的正性保持可推导出不变域保持特性。为避免伪振荡，我们额外对密度、速度分量及比总能中间状态施加局部极值原理。针对后向欧拉时间离散，通过构建具有不变域保持特性且在强时间步长限制下收敛的不动点迭代，证明了全离散MCL格式的不变域保持（IDP）性质。针对非线性离散问题的迭代求解器采用更高效的不动点迭代法，其关联线性系统的矩阵采用鲁棒性强的低阶雅可比近似，该近似利用了通量函数的齐次性。限制性反扩散项采用显式处理。我们以正性保持作为非线性迭代的终止准则：首次迭代得到线性化半隐式问题的解，该解具备离散守恒特性但通常不满足IDP性质；若检测到非IDP状态则继续迭代。理论分析保证了IDP极限解的存在性。为促进稳态解的收敛，我们在每个时间步末执行自适应显式欠松弛处理，其松弛因子的计算基于节点熵残差的近似最小化。通过典型二维测试问题的收敛历程，展示了所提算法与替代求解策略的性能表现。