[Paper Review] A quasi-conservative DG-ALE method for multi-component flows using the non-oscillatory kinetic flux

Dongmi Luo, Shiyi Li+3 more|arXiv (Cornell University)|Jan 13, 2021

Computational Fluid Dynamics and Aerodynamics52 references4 citations

TL;DR

This paper proposes a high-order quasi-conservative discontinuous Galerkin arbitrary Lagrangian-Eulerian (DG-ALE) method for compressible multi-component flows using a non-oscillatory kinetic flux. The method employs a predictor-corrector grid velocity strategy—first computing a Lagrangian mesh based on flow velocity, then improving its quality via the MMPDE moving mesh method—enabling accurate, high-resolution resolution of shocks and material interfaces with minimal oscillations.

ABSTRACT

A high-order quasi-conservative discontinuous Galerkin (DG) method is proposed for the numerical simulation of compressible multi-component flows. A distinct feature of the method is a predictor-corrector strategy to define the grid velocity. A Lagrangian mesh is first computed based on the flow velocity and then used as an initial mesh in a moving mesh method (the moving mesh partial differential equation or MMPDE method ) to improve its quality. The fluid dynamic equations are discretized in the direct arbitrary Lagrangian-Eulerian framework using DG elements and the non-oscillatory kinetic flux while the species equation is discretized using a quasi-conservative DG scheme to avoid numerical oscillations near material interfaces. A selection of one- and two-dimensional examples are presented to verify the convergence order and the constant-pressure-velocity preservation property of the method. They also demonstrate that the incorporation of the Lagrangian meshing with the MMPDE moving mesh method works well to concentrate mesh points in regions of shocks and material interfaces.

Motivation & Objective

• To develop a high-order numerical method for compressible multi-component flows that accurately resolves complex features such as shocks, contact discontinuities, and material interfaces.
• To overcome mesh distortion in pure Lagrangian methods by combining Lagrangian meshing with a moving mesh optimization strategy.
• To maintain conservation properties and avoid spurious oscillations near material interfaces using a quasi-conservative DG formulation for species equations.
• To achieve high-order accuracy and robustness in both smooth and discontinuous flow regimes through a direct ALE framework with adaptive meshing.
• To demonstrate the method’s ability to concentrate mesh points naturally at physical discontinuities without remapping or remeshing artifacts.

Proposed method

• A predictor-corrector strategy is used to define the grid velocity: a Lagrangian mesh is first predicted based on the flow velocity.
• The MMPDE (moving mesh partial differential equation) method is applied to correct the initial Lagrangian mesh, improving its quality while preserving concentration at shocks and interfaces.
• The fluid dynamics equations are discretized using DG elements in the direct ALE framework with the non-oscillatory kinetic flux to avoid Riemann solver construction.
• Species transport equations are solved using a quasi-conservative DG scheme to suppress numerical oscillations near material interfaces.
• The method supports both $P^1$ and $P^2$ discontinuous Galerkin elements for high-order accuracy in space.
• The framework is implemented in a fully conservative and high-order manner, with time integration handled via Runge-Kutta schemes.

Experimental results

Research questions

• RQ1 Can a high-order DG-ALE method with a non-oscillatory kinetic flux achieve accurate resolution of shocks and material interfaces in multi-component compressible flows?
• RQ2 Does combining Lagrangian meshing with the MMPDE moving mesh method effectively maintain mesh quality while concentrating points at discontinuities?
• RQ3 Can the quasi-conservative DG formulation prevent spurious oscillations in species transport near material interfaces?
• RQ4 How does the method perform in terms of convergence order and preservation of constant pressure-velocity states in benchmark problems?
• RQ5 To what extent does the method maintain accuracy and robustness under high-pressure ratio conditions, such as in underwater explosion scenarios?

Key findings

• The method achieves optimal $k+1$ order of accuracy for $P^k$-DG elements in smooth flow problems, confirming theoretical convergence rates.
• The constant-pressure-velocity preservation property is numerically verified, indicating the method maintains equilibrium states without spurious perturbations.
• Mesh concentration at material interfaces and shock waves is effectively achieved, with $P^2$-DG producing sharper density contours than $P^1$-DG.
• In the underwater explosion test case, the method successfully captures the outward shock wave in water, inward rarefaction in gas, and interface deformation from circular to oval-like shape.
• The combination of Lagrangian meshing and MMPDE correction results in high-quality, well-conditioned meshes that track discontinuities without tangling or excessive distortion.
• The non-oscillatory kinetic flux effectively avoids the need for Riemann solvers while maintaining robustness and accuracy in multi-component flows.