University of Kansas: Smith Colloquium Fall 2025
On October, 23, 2025, Professor Eduardo Abreu will be presenting in the Smith Colloquium Fall 2025 the following work:
On the Lagrangian-Eulerian Approach: Mathematical and Computational Aspects with Examples
Mathematical modeling and numerical analysis challenges for the study of hyperbolic partial differential equations (PDEs) is in the realm of basic and applied sciences, for instance, ranging from fluid mechanics and modeling of vehicular traffic flows to fluid dynamics in porous media flows. In this Colloquium, we will discuss on a new approach [1,2,3] for studying some hyperbolic conservation laws of first order PDEs with examples for some multi-dimensional problems subject to irregular vector fields either in fluid mechanics linked to vortex sheet [4] or in the context of flow is porous media with spatially discontinuous coefficients [5]. In the context of multidimensional hyperbolic systems of conservation laws, the resulting Lagrangian-Eulerian method [6] satisfies a weak positivity principle in view of results of P. Lax and X.-D. Liu [Computational Fluid Dynamics Journal, 5(2) (1996) 133-156 and [Journal of Computational Physics, 187 (2003) 428-440]. We also found [2] an interesting connection between the notion of no-flow curves [1,2,3] (viewed as a vector field with locally bounded variation) and the results of A. Bressan in the context of (local) existence and continuous de pendence for discontinuous O.D.E.’s as introduced by A. Bressan (1988) [Proc. Amer. Math. Soc. 104, 772-778]. The method is based on the concept of multidimensional no-flow curves/surfaces/manifolds [1,2,3,6]. Roughly speaking, one reduces the hyperbolic PDE into a family of ODEs along the forward untangled space-time no-flow Lagrangian trajectories. As a by-product of the no-flow framework, there is no need to compute the eigenvalues (exact or approximate values), and in fact there is no need to construct the Jacobian matrix of the hyperbolic flux functions, and thus giving rise to an effective (weak) CFL-stability condition useful in the computing practice. The no-flow framework might be also applied to nonlinear balance laws [3]. We present numerical computations for nontrivial (local and nonlocal) hyperbolic problems, as such compressible Euler flows with positivity of the density, the Orszag-Tang problem, which is well-known to satisfy the notable involution-constrained partial differential equation div B = 0, a nonstrictly hyperbolic three-phase flow system in porous media with a resonance point, and the classical 3 by 3 shallow-water system (with and without discontinuous bottom topography). We will also provide numerical 1-D and Multi-D examples to verify the theory and exemplify the capabilities of the proposed approach. E. Abreu thanks the grant support of CNPq (307641/2023-6) and FAPESP (2025/07662-6).
For more information on the Smith Colloquium Fall 2025: https://mathematics.ku.edu/smith-colloquium-fall-2025
To access the abstracts to all presentations: https://mathematics.ku.edu/colloquium-abstracts-fall-2025