Active flux methods for hyperbolic conservation laws -- flux vector splitting and bound-preservation (2411.00065v3)

Abstract: The active flux (AF) method is a compact high-order finite volume method that simultaneously evolves cell averages and point values at cell interfaces. Within the method of lines framework, the existing Jacobian splitting-based point value update incorporates the upwind idea but suffers from a stagnation issue for nonlinear problems due to inaccurate estimation of the upwind direction, and also from a mesh alignment issue partially resulting from decoupled point value updates. This paper proposes to use flux vector splitting for the point value update, offering a natural and uniform remedy to those two issues. To improve robustness, this paper also develops bound-preserving (BP) AF methods for hyperbolic conservation laws. Two cases are considered: preservation of the maximum principle for the scalar case, and preservation of positive density and pressure for the compressible Euler equations. The update of the cell average is rewritten as a convex combination of the original high-order fluxes and robust low-order (local Lax-Friedrichs or Rusanov) fluxes, and the desired bounds are enforced by choosing the right amount of low-order fluxes. A similar blending strategy is used for the point value update. In addition, a shock sensor-based limiting is proposed to enhance the convex limiting for the cell average, which can suppress oscillations well. Several challenging tests are conducted to verify the robustness and effectiveness of the BP AF methods, including flow past a forward-facing step and high Mach number jets.

• The paper introduces flux vector splitting in active flux methods, significantly improving shock capturing and stability over traditional Jacobian splitting.
• The paper presents a convex blending strategy that enforces bound-preserving limits to maintain physical consistency in compressible flow simulations.
• The paper implements a novel shock sensor for adaptive dissipation control, enabling accurate resolution of discontinuities in high-speed flows.

Overview of Active Flux Methods with Flux Vector Splitting

The paper presents advancements in the domain of active flux (AF) methods applied to hyperbolic conservation laws by proposing the use of flux vector splitting (FVS) to improve the robustness, accuracy, and stability of numerical solutions. Focused primarily on the domains of fluid dynamics governed by hyperbolic models such as the Euler equations, these methods aim to refine the treatment of nonlinear problems and high-speed flows.

The paper identifies key limitations in the existing AF methods, particularly those based on Jacobian splitting, which tend to suffer from stagnation and mesh alignment issues affecting convergence and stability. The proposed solution leverages FVS to naturally and uniformly address these challenges. Furthermore, the research extends the bounded solution space by incorporating a bound-preserving (BP) strategy to handle scalar cases with maximum principles and ensure positive density and pressure in compressible flows, especially for the Euler equations.

Key Contributions

1. Flux Vector Splitting in Point Value Updates: The transition from Jacobian splitting to FVS establishes a robust framework mitigating stagnation and alignment problems in AF methods. This strategic shift is shown to enhance the capturing of shock waves and accommodate larger CFL conditions, particularly when implemented with LLF-based FVS.
2. Bound-Preserving Strategies: The paper introduces a convex limiting technique for cell averages combining higher-order and robust low-order flux schemes. This convex blending guarantees the numerical solution remains within predefined bounds, thus securing the physical feasibility of AI-generated solutions across grid scales.
3. Shock Sensor-Based Limiting: To counteract oscillations around shock discontinuities, a novel shock sensor method is proposed. Balancing between accuracy and dissipation, the sensor provides an adaptive mechanism enhancing the fidelity of flow simulations especially in scenarios involving complex, small-scale turbulence.

Numerical Validation

Through rigorous simulation experiments, the proposed methods exhibit enhanced performance under challenging test scenarios including self-steepening shock scenarios in Burgers' equations, the LeBlanc shock tube, and high Mach number jet flows, among others. Crucially, the AF methods using FVS reveal better stability and sharper interface capturing compared to traditional techniques, demonstrated vividly in the successful handling of near-vacuum states and blast wave interactions.

Implications and Potential Developments

The implications of this paper stretch across theoretical and practical domains. Theoretically, the introduction of FVS into the AF method enriches the framework, paving the way for further exploration in non-linear hyperbolic PDEs and related systems. Practically, this approach sees potential applications in high-speed aerodynamic computations, astrophysical simulations, and beyond, where the accuracy and stability of shockwave interactions are critical.

Looking forward, the modular nature of test cases suggests opportunities for extending these methods to multi-dimensional, multi-physics scenarios in computational fluid dynamics. Additionally, the intrinsic adaptability of these numerical strategies could guide future advances in data-driven simulations governed by AI-augmented methods.

In conclusion, by addressing long-standing numerical challenges within hyperbolic conservation laws, this paper enriches the toolkit available to researchers and practitioners, marking a firm step toward more consistent and reliable granularity in fluid mechanics modeling.