Well-Balanced Finite-Volume Schemes for Hydrodynamic Equations with General Free Energy

High-Order Well-Balanced Finite-Volume Schemes for Hydrodynamic Equations With Nonlocal Free Energy

“…In our previous work [20] we extended the applicability of well-balanced schemes to the broad class of hydrodynamic models with attractive-repulsive interaction forces. In particular, we considered interactions associated with nonlocal convolutions or functions of convolutions, which is commonplace in applications such as the Keller-Segel model [9], more general Euler-Poisson systems [55], or in dynamic-density functional theory (DDFT) [47,48].…”

“…First, in section 2 we explain the construction of our well-balanced high-order finite-volume scheme. In subsection 2.1 we begin by recalling our first-order numerical scheme from [20], and then in subsection 2.2 we provide a first-attempt to extend such a scheme to high order. The correct wellbalanced formulation for that high-order scheme is provided in subsection 2.3.…”

“…In what follows we begin by briefly recalling in subsection 2.1 the first-order well-balanced scheme for (2.1) introduced in [20], which serves as a starting point to construct high-order schemes by employing a high-order reconstruction operator, as described in subsection 2.2. Then in subsection 2.3 we describe how to adapt these high-order schemes so that they are well-balanced in the sense defined in 2.1.…”

“…First-order numerical scheme. The first-order semidiscrete well-balanced finite-volume scheme for system (2.1) introduced in [20] can be written as (2.5)…”

“…Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/page/terms A833 well as its well-balanced character. In [20] the midpoint quadrature formula is used to approximate both the cell averages of the exact solution and H ∆x (x). In what follows we suppress the time dependence on the cell averages for simplicity.…”