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We propose novel less diffusive schemes for conservative one- and two-dimensional hyperbolic systems of nonlinear partial differential equations (PDEs). The main challenges in the development of accurate and robust numerical methods for the…

Numerical Analysis · Mathematics 2022-11-09 Alina Chertock , Shaoshuai Chu , Michael Herty , Alexander Kurganov , Maria Lukacova-Medvidova

Simulating the flow of two fluid phases in porous media is a challenging task, especially when fractures are included in the simulation. Fractures may have highly heterogeneous properties compared to the surrounding rock matrix,…

Numerical Analysis · Mathematics 2024-07-09 Enrico Ballini , Luca Formaggia , Alessio Fumagalli , Eirik Keilegavlen , Anna Scotti

In this paper a fully Eulerian solver for the study of multiphase flows for simulating the propagation of surface gravity waves over submerged bodies is presented. We solve the incompressible Navier-Stokes equations coupled with the volume…

Fluid Dynamics · Physics 2021-06-02 Francesco De Vita , Filippo De Lillo , Roberto Verzicco , Miguel Onorato

We consider admissible weak solutions to the compressible Euler system with source terms, which include rotating shallow water system and the Euler system with damping as special examples. In the case of anti-symmetric sources such as…

Analysis of PDEs · Mathematics 2015-06-04 Tianwen Luo , Chunjing Xie , Zhouping Xin

We study dissipative weak (DW) solutions of the Euler equations of gas dynamics using the first-, second-, third-, fifth-, seventh-, and ninth-order local characteristic decomposition-based central-upwind (LCDCU), low-dissipation…

Numerical Analysis · Mathematics 2026-01-28 Shaoshuai Chu , Michael Herty , Alexander Kurganov , Maria Lukacova-Medvidova , Changsheng Yu

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…

Numerical Analysis · Mathematics 2025-03-21 Junming Duan , Wasilij Barsukow , Christian Klingenberg

Numerical simulation of multi-component flow systems characterized by the simultaneous presence of pressure-velocity coupling and pressure-density coupling dominated regions remains a significant challenge in computational fluid dynamics.…

Fluid Dynamics · Physics 2025-10-17 Xi Deng , Bin Xie , Omar K. Matar , Pierre Boivin

We present a convergence analysis of a finite volume (FV) scheme for the multicomponent compressible Euler system in the framework of dissipative weak (DW) solutions. DW solutions were introduced as a generalized solution framework in…

Numerical Analysis · Mathematics 2026-05-26 Jaya Agnihotri , Philipp Öffner

We present a new method for numerical hydrodynamics which uses a multidimensional generalisation of the Roe solver and operates on an unstructured triangular mesh. The main advantage over traditional methods based on Riemann solvers, which…

Instrumentation and Methods for Astrophysics · Physics 2017-06-28 Sijme-Jan Paardekooper

In this work, we introduce a framework to design multidimensional Riemann solvers for nonlinear systems of hyperbolic conservation laws on general unstructured polygonal Voronoi-like tessellations. In this framework we propose two simple…

Numerical Analysis · Mathematics 2026-02-03 Elena Gaburro , Mario Ricchiuto , Michael Dumbser

Eulerian hydrodynamical simulations are a powerful and popular tool for modeling fluids in astrophysical systems. In this work, we critically examine recent claims that these methods violate Galilean invariance of the Euler equations. We…

Cosmology and Nongalactic Astrophysics · Physics 2011-06-03 Brant E. Robertson , Andrey V. Kravtsov , Nickolay Y. Gnedin , Tom Abel , Douglas H. Rudd

Changes of type transitions for the two-component hydrodynamic type systems are discussed. It is shown that these systems generically assume the Jordan form (with 2 X 2 Jordan block) on the transition line with hodograph equations becoming…

Mathematical Physics · Physics 2015-10-12 B. G. Konopelchenko , G. Ortenzi

Split form schemes for Euler and Navier-Stokes equations are useful for computation of turbulent flows due to their better robustness. This is because they satisfy additional conservation properties of the governing equations like kinetic…

Numerical Analysis · Mathematics 2021-05-03 Vikram Singh , Praveen Chandrashekar

The Lagrange-Flux schemes are Eulerian finite volume schemes that make use of an approximate Riemann solver in Lagrangian description with particular upwind convective fluxes. They have been recently designed as variant formulations of…

Numerical Analysis · Mathematics 2017-03-02 Florian De Vuyst

The accurate numerical simulation of high Reynolds number incompressible flows is a challenging topic in computational fluid dynamics. Classical inf-sup stable methods like the Taylor-Hood element or only $L^2$-conforming discontinuous…

Numerical Analysis · Mathematics 2019-12-24 Marian Piatkowski , Peter Bastian

We prove via convex integration a result that allows to pass from a so-called subsolution of the isentropic Euler equations (in space dimension at least $2$) to exact weak solutions. The method is closely related to the incompressible…

Analysis of PDEs · Mathematics 2021-07-23 Tomasz Dębiec , Jack W. D. Skipper , Emil Wiedemann

This article provides a reduced-order modelling framework for turbulent compressible flows discretized by the use of finite volume approaches. The basic idea behind this work is the construction of a reduced-order model capable of providing…

Fluid Dynamics · Physics 2024-05-31 Matteo Zancanaro , Valentin Nkana Ngan , Giovanni Stabile , Gianluigi Rozza

The flux vector splitting (FVS) method has firstly been incorporated into the discontinuous Galerkin (DG) framework for reconstructing the numerical fluxes required for the spatial semi-discrete formulation, setting it apart from the…

Numerical Analysis · Mathematics 2024-12-12 Zhengrong Xie

The present paper addresses the development and implementation of the first high-order Flux Reconstruction (FR) solver for high-speed flows within the open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid Dynamics)…

Computational Physics · Physics 2019-07-24 Ray Vandenhoeck , Andrea Lani

In this paper, we study the well-posedeness at low regularity of a two-dimensional system obtained as a reduced model for micropolar fluid dynamics. At the mathematical level, the system presents a coupling between an Euler-type equation…

Analysis of PDEs · Mathematics 2026-05-14 Francesco Fanelli , Pedro Gabriel Fernández Dalgo