[Paper Review] A robust conservative mixed finite element method for compressible flow on pipe networks

Herbert Egger|arXiv (Cornell University)|Sep 16, 2016

Advanced Numerical Methods in Computational Mathematics12 references3 citations

TL;DR

This paper presents a robust, conservative mixed finite element method for simulating compressible flow in pipe networks using a variational formulation that enforces global conservation of mass and energy. The method employs conforming Galerkin discretization in space and an implicit time-stepping scheme that preserves mass exactly and dissipates energy slightly, ensuring well-posedness and stability across diverse network topologies and parameters.

ABSTRACT

We consider the numerical approximation of compressible flow in a pipe network. Appropriate coupling conditions are formulated that allow us to derive a variational characterization of solutions and to prove global balance laws for the conservation of mass and energy on the whole network. This variational principle, which is the basis of our further investigations, is amenable to a conforming Galerkin approximation by mixed finite elements. The resulting semi-discrete problems are well-posed and automatically inherit the global conservation laws for mass and energy from the continuous level. We also consider the subsequent discretization in time by a problem adapted implicit time stepping scheme which leads to conservation of mass and a slight dissipation of energy of the full discretization. The well-posedness of the fully discrete scheme is established and a fixed-point iteration is proposed for the solution of the nonlinear systems arising in every single time step. Some computational results are presented for illustration of our theoretical findings and for demonstration of the robustness and accuracy of the new method.

Motivation & Objective

• To develop a numerical method that conserves mass and energy globally in compressible flow on pipe networks.
• To derive coupling conditions at junctions that ensure physical consistency and conservation of mass and energy.
• To formulate a variational principle that naturally encodes conservation laws and enables conforming Galerkin discretization.
• To establish well-posedness and conservation properties for both semi-discrete and fully discrete schemes.
• To demonstrate robustness and accuracy through numerical experiments on Riemann-type flow problems and steady-state convergence.

Proposed method

• A variational formulation is derived from the governing equations of compressible flow, incorporating mass and momentum balance with a pressure law $ p = c\rho^\gamma $.
• Coupling conditions at pipe junctions are formulated to enforce conservation of mass flux and stagnation enthalpy, including viscous effects.
• A mixed finite element method is applied for spatial discretization, ensuring conforming Galerkin approximation and exact mass conservation at the semi-discrete level.
• An implicit time-stepping scheme is used that preserves mass exactly and introduces controlled numerical dissipation of energy.
• Nonlinear systems arising at each time step are solved via a fixed-point iteration, ensuring robustness and convergence.
• Energy estimates and stability analysis are used to establish well-posedness of the fully discrete scheme.

Experimental results

Research questions

• RQ1 Can a variational formulation be derived that inherently encodes global conservation of mass and energy in compressible flow on pipe networks?
• RQ2 How can coupling conditions at junctions be formulated to ensure physical consistency and conservation of mass and energy?
• RQ3 Does a conforming mixed finite element method applied to the variational formulation preserve mass and energy conservation at the discrete level?
• RQ4 Can an implicit time discretization be designed to conserve mass and introduce only controlled energy dissipation?
• RQ5 How robust is the fully discrete scheme with respect to model parameters, mesh refinement, and initial conditions?

Key findings

• The proposed method exactly conserves mass at the discrete level, which is crucial for achieving correct steady-state solutions.
• Energy is slightly dissipated by the implicit time discretization, mimicking physical viscous dissipation and ensuring stability.
• The semi-discrete and fully discrete schemes are well-posed, with convergence observed under mesh and time step refinement.
• Numerical experiments show robust behavior across different initial conditions, including discontinuous density profiles, with rapid smoothing of discontinuities due to numerical damping.
• The method maintains conservation properties and stability even for small viscosity parameters and complex network topologies.
• The fixed-point iteration converges reliably for all test cases, demonstrating the robustness of the nonlinear solver.