Overview of Current Work & Advanced Computing
This document presents the ongoing research, simulations, and numerical frameworks currently being developed. The primary focus of this stage is the implementation of a no-flow Lagrangian-Eulerian framework, which is being validated and refined through various benchmark problems based on classical and novel mathematical models. Currently we are focusing in the following models:
The ongoing simulations and core research directions are structured around three fundamental pillars:
1: Forward-Facing Step Problem (Mach 3 Wind Tunnel)
Context: This benchmark simulation addresses high-speed aerodynamic flows within a wind tunnel configuration.
Theoretical Foundation: This work is directly inspired by and builds upon the methodology established by Lukácová-Medvid'ová and Rohde (2024).
2: Isentropic Compressible Euler Equations
Context: Investigation into the behavior of compressible fluids under isentropic conditions, focusing on wave patterns and structural stability.
Theoretical Foundation: The mathematical formulations and analytical approaches are guided by the seminal works of Bressan and Murray (2020), as well as Bressan, Jiang, and Liu (2021).
Filled Origin (epsilon=0.001)
Hole on Origin (epsilon=0.001)
Filled Origin (epsilon=0.004)
Hole on Origin (epsilon=0.004)
Filled Origin (epsilon=0.008)
Hole on Origin (epsilon=0.008)
Hole on Origin (epsilon=0.008) Extended
3: Relaxed Navier-Stokes Equations (Lid-driven Cavity)
Context: Application of a fluid dynamics benchmark (the lid-driven cavity) to test dissipative and viscous behaviors under a relaxed framework.
Theoretical Foundation: This segment implements a novel hyperbolic relaxation for the Navier-Stokes Equations proposed by Huang, Rohde, Yong, and Zhang (2025).
Note to the Reader: For a comprehensive understanding of the precise hyperbolic relaxation mathematical equations being solved in the lid-driven cavity simulation, please refer directly to the original paper by Huang et al. (2025).
Re=1000
Re=5000
Re=10000
Re=20000
Re=30000
Mathematical Foundations and Advanced Computing of No-Flow Curves:
All four below simulations are structurally embedded within the mathematical framework of "A vector field approach to the evolution and remapping of hyperbolic fluxes on cubical and tetrahedral meshes". This framework reinterprets the evolution of hyperbolic fluxes as a discontinuous space-time dependent vector field problem with locally bounded cone-variation, utilizing the theory of discontinuous Ordinary Differential Equations (ODEs) by Bressan and Colombo. By avoiding Riemann solvers, Jacobian constructions, and spectral information, the framework guarantees a weak CFL-type stability and natural system positivity during the dual-step evolution-remap cycle.
Theme I: Mathematical Visualization of No-Flow Curves
1. Simulation Context: Diagonal/Side Perspective
This simulation presents a 3D diagonal and side perspective, visualized via Polyscope, of the spatial trajectories generated by the No-Flow Curves during the evolution of a three-dimensional Burgers-type flux interacting with a localized disk discontinuity. The primary objective of this view is to visually validate the geometric existence and continuous dependence of solutions under the Bressan-Colombo framework for discontinuous ODEs. The isometric perspective highlights the non-trivial bending, twisting, and topological behavior of the trajectories as they cross discontinuous interfaces, illustrating how the vector field behaves under locally bounded cone-variation without suffering from numerical oscillations.
2. Simulation Context: Top/Orthogonal Perspective
Rendered from an orthogonal top-down perspective, this simulation isolates the projection of the 3D Lagrangian-Eulerian (LE) No-Flow Curves onto a two-dimensional planar cross-section. Captured at a precise execution timestamp, this visualization maps out the exact geometric paths along which the hyperbolic fluxes evolve prior to the remapping phase. This clean top view allows for a rigorous inspection of fluid boundaries and contact deformations. It visually verifies that the weak CFL-type stability conditions are satisfied locally, ensuring that intersecting trajectories do not violate entropy conditions or compromise natural system positivity.
Theme II: Mathematical Exploration for Advanced Algorithm Design (2D & 3D)
3. Simulation Context: Mesh Interaction (Diagonal/Top Perspective)
This simulation explores the algorithmic mapping of the abstract No-Flow framework onto explicit multi-dimensional computational meshes. Viewed from a top-diagonal perspective within Polyscope, this simulation demonstrates the dual-step evolution-remap strategy acting upon a Burgers-disk benchmark. It specifically illustrates how the continuous vector field lines interact with and cross the boundaries of structured cubical elements and tetrahedral transitions. By translating these abstract manifolds into discrete cell-boundary flux updates, the simulation demonstrates the feasibility of designing advanced, Riemann-solver-free algorithms that completely bypass Jacobian matrix constructions across complex, multi-element mesh interfaces.
4. Simulation Context: Mesh Alignment (Top Perspective)
This simulation provides a precise, top-down orthogonal overview of the mesh-bound No-Flow evolution scheme. Captured synchronously with the core algorithmic updates, this view details how the flux transport vectors are distributed across the cellular elements of the mesh. This 3D-to-2D projection is highly critical for verifying the structural preservation properties of the algorithm, such as maintaining the divergence-free condition (∇⋅B=0) in complex magnetohydrodynamics (MHD) problems or ensuring robustness near resonance points in non-strictly hyperbolic three-phase flows. The alignment between the grid lines and the mathematical curves confirms the accuracy and efficiency of the multi-dimensional discretization framework.
Additional results from our work:
Buckley-Leverett equation with gravity
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Three-phase flow model in porous media
Model 1: Three-phase Buckley-Leverett shock wave followed by composite wave comprising a shock wave and adjoining rarefaction, that is, there is no occurrence of the non-classical transitional shock.
Model 2: Three-phase Buckley-Leverett problem: occurrence of the non-classical transitional shock.
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2D compressible Euler equations for several problems
Model 1: Case involving slip lines (Riemann Problem I)
Model 2: Case involving slip lines (Riemann Problem II)
Model 3: Double Mach reflection (Density)
Model 4: Double Mach reflection (Pressure)
Model 5: A Mach 3 wind tunnel with a step (Density)
Model 6: A Mach 3 wind tunnel with a step (Pressure)
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2D shallow water system
With non-flat bottom (left) and with discontinuous topography (right)
2D shallow water system discontinuous bottom topography
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2D Orszag-Tang Vortex (in closed form)
2D Orszag-Tang Vortex (in closed form)
Or in a suitable (conservative) open form:
Density
Pressure
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A 2D nonlinear pseudo-parabolic Buckley-Leverett saturation transport model coupled with an elliptic pressure-velocity problem
Test Case 3) Saturation for a 2D nonlinear pseudo-parabolic Buckley-Leverett Profile during the infiltration experiment: Homogeneous porous media
Test Case 1a) 3D VIEW: Numerical resolution of a 2D nonlinear pseudo-parabolic Buckley-Leverett saturation transport model coupled with an elliptic pressure-velocity problem - Viscosity ratio effects
Test Case 1b) 2D COLOR MAP VIEW: Numerical resolution of a 2D nonlinear pseudo-parabolic Buckley-Leverett saturation transport model coupled with an elliptic pressure-velocity problem - Viscosity ratio effects
Test Case 4) Saturation for a 2D nonlinear pseudo-parabolic Buckley-Leverett Profile during the infiltration experiment: Heterogeneous porous media