Conservation Laws with Discontinuous Flux Functions
Summary
Conservation laws with discontinuous flux functions govern a wide range of phenomena from fluid flow in heterogeneous porous media to traffic dynamics on road networks. These laws take the form ∂ₜu + ∂ₓf(x,u)=0, where the flux function f may change abruptly in space or time due to varying material properties or imposed constraints. Discontinuities in the flux can induce complex solution structures, including stationary shocks, non-classical waves at interfaces and regions of mixed hyperbolic–parabolic behaviour when regularised by small viscosity terms. The theoretical analysis centres on establishing well-posedness, typically via the introduction of entropy conditions that rule out non-physical oscillations, and on proving compactness and convergence for approximation schemes. Numerically, special finite volume, finite difference and spectral methods are developed to capture sharp transitions without spurious oscillations. Practical applications span petroleum engineering, where fluid interfaces in layered reservoirs produce discontinuous flow laws, to the control of traffic congestion through moving bottlenecks and junction models. Current research seeks to unify existence and uniqueness arguments across heterogeneous domains, refine high-order accurate schemes that preserve equilibrium states at flux jumps, and sharpen quantitative estimates on solution regularity and error bounds in both deterministic and stochastic settings.
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Conservation Laws with Discontinuous Flux Functions publication trend
The graph below shows the total number of articles in conservation laws with discontinuous flux functions across all publications each year (not limited to Nature Index journals).
Technical terms
Conservation law: A partial differential equation expressing the invariance of a physical quantity over time, typically in the form ∂ₜu + ∂ₓf(x,u)=0.
Flux function: A mapping f(x,u) that defines the rate at which the conserved quantity u is transported, which may vary in space or exhibit discontinuities.
Discontinuous flux: A flux function that changes abruptly at interfaces, modelling heterogeneous media or imposed constraints and inducing non-classical wave interactions.
Entropy solution: A weak solution satisfying additional inequalities (entropy conditions) that ensure uniqueness and physical admissibility by excluding spurious oscillations.
Vanishing viscosity: A regularisation technique adding a small diffusive term ε∂ₓₓu to the conservation law, used to select physically relevant entropy solutions in the limit ε→0.
References
- Compactness estimates for difference schemes for conservation laws with discontinuous flux. IMA Journal of Numerical Analysis (2024).
- A filtered Chebyshev spectral method for conservation laws on network. Computers & Mathematics with Applications (2023).
- A LWR model with constraints at moving interfaces. ESAIM Mathematical Modelling and Numerical Analysis (2022).
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