A first-order hyperbolic system approach for dispersion

Mazaheri, Alireza; Ricchiuto, Mario; Nishikawa, Hiroaki

doi:10.1016/j.jcp.2016.06.001

Cited by 29 publications

(25 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Bodies or boundaries present in the domain are described implicitly in the mesh by means of a distance function, or 1. We elaborate on the enrichment approach used in [35,31,32,33] in the context of body fitted stabilized finite elements in mixed form. We propose a symmetric enrichment method that can be rigorously proved to be stable and third-/second-order accurate for the primary variable/gradient, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…In [28], for the Laplace problem, the use of a reconstructed gradient has been used to overcome this difficulty.A more robust strategy to achieve accurate gradients is pursued here, and consists in developing a mixed formulation, in which the Poisson operator is replaced by the Darcy operator, and to design advanced schemes for high-order flux approximations. In this context, we would like to mention recent contributions [35,31,32,33] for body-fitted computations, which were inspired by the earlier work of Caraeni [9]. In these works, mixed forms of the Poisson, diffusion, or advection diffusion operators were proposed, in which the solution derivatives are adopted as main unknowns, together with the solution itself.…”

mentioning

confidence: 99%

See 1 more Smart Citation

High-order gradients with the shifted boundary method: An embedded enriched mixed formulation for elliptic PDEs

Nouveau

Ricchiuto

Scovazzi

2019

Journal of Computational Physics

View full text Add to dashboard Cite

We propose an extension of the embedded boundary method known as "shifted boundary method" to elliptic diffusion equations in mixed form (e.g., Darcy flow, heat diffusion problems with rough coefficients, etc.). Our aim is to obtain an improved formulation that, for linear finite elements, is at least second-order accurate for both flux and primary variable, when either Dirichlet or Neumann boundary conditions are applied. Following previous work of Nishikawa and Mazaheri in the context of residual distribution methods, we consider the mixed form of the diffusion equation (i.e., with Darcy-type operators), and introduce an enrichment of the primary variable. This enrichment is obtained exploiting the relation between the primary variable and the flux variable, which is explicitly available at nodes in the mixed formulation. The proposed enrichment mimics a formally quadratic pressure approximation, although only nodal unknowns are stored, similar to a linear finite element approximation. We consider both continuous and discontinuous finite element approximations and present two approaches: a non-symmetric enrichment, which, as in the original references, only improves the consistency of the overall method; and a symmetric enrichment, which enables a full error analysis in the classical finite element context. Combined with the shifted boundary method, these two approaches are extended to high-order embedded computations, and enable the approximation of both primary and flux (gradient) variables with second-order accuracy, independently on the type of boundary conditions applied. We also show that the the primary variable is third-order accurate, when pure Dirichlet boundary conditions are embedded. similar indicator function. The main idea behind the IBM is to solve the model equations on the entire domain, and to impose the boundary conditionss via a forcing term. The method was originally designed on Cartesian grids and its development has been an active field of research for some decades now. Two exhaustive reviews were written by Mittal and Iaccarino in 2005 [34], and Sotiropoulos and Yang in 2014 [47]. The main drawback of the IBM is the accuracy of boundary conditions, which are most often only first-order accurate. Some strategies have been proposed in the past to increase the IBM accuracy, at the expense of more involved implementations [18,17,26,27], or to compensate low-order accuracy by means of mesh adaptation [38,6,19].A similar objective is pursued by Embedded Boundary Methods, which still use an immersed description of bodies and boundaries, but solve the model equations only in the regions of interest. Within a finite element context, the cut-cell and cutFEM approach [39,15,20,10,24,48] are most commonly utilized. These techniques usually combine a weak enforcement of the boundary conditions with a XFEM strategy. For example, in CutFEMs, the boundary is reconstructed as the intersection between the boundary surface and the embedding (underlaying) grid. Approaches of this kind often require complex...

show abstract

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

High-order gradients with the shifted boundary method: An embedded enriched mixed formulation for elliptic PDEs

Nouveau

Ricchiuto

Scovazzi

2019

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…The effect of nonhydrostatic distributions of pressure is used in these models by using additional 'internal' variables in the equations. In [15,26] hyperbolic techniques are applied to dispersive systems in a similar way. The main advantage of such hyperbolic approximation of the dispersive equations is essential simplification of the algorithms of numerical calculation and formulation of the boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic model for free surface shallow water flows with effects of dispersion, vorticity and topography

Чесноков

Nguyen

2019

Computers & Fluids

View full text Add to dashboard Cite

We derive a hyperbolic system of equations approximating the two-layer dispersive shallow water model for shear flows recently proposed by Gavrilyuk, Liapidevskii & Chesnokov (J. Fluid Mech., vol. 808, 2016, pp. 441-468). The use of this system for modelling the evolution of surface waves makes it possible to avoid the major numerical challenges in solving dispersive shallow water equations, which are connected with the resolution of an elliptic problem at each time instant and realization of non-reflecting conditions at the boundary of the calculation domain. It also allows one to reduce the computation time. The velocities of the characteristics of the obtained model are determined and the linear analysis is performed. Stationary solutions of the model are constructed and studied. Numerical solutions of the hyperbolic system are compared with solutions of the original dispersive model. It is shown that they almost coincide for large time intervals. The system obtained is applied to study non-stationary undular bores produced after interaction of a uniform flow with an immobile wall, non-hydrostatic shear flows over a local obstacle and the evolution of breaking solitary wave on a sloping beach.

show abstract

“…The mathematical strategy of this approach is to split the second order partial differential equation into a set of first-order differential equations by adding new variables and pseudo-time advancement terms such that the diffusion equation is reformulated as a hyperbolic system. This radical approach has been shown to offer several advantages over conventional methods, such as accelerated convergence for steady state solution and higher order of accuracy for both primary and gradient variables, as demonstrated for diffusion [16], the incompressible/compressible Navier-Stokes equations [17,18], third-order dispersion equations [19], an incompressible magnetohydrodynamics model [20], an elliptic distance-function model [21], and so on. The original approach of Nishikawa has been further extended to a constant diffusion tensor by Lou et al [22,23] discretized with the reconstructed discontinuous Galerkin scheme (rDG).…”

Section: Introductionmentioning

confidence: 99%

First order hyperbolic approach for Anisotropic Diffusion equation

Chamarthi

Nishikawa

Komurasaki

2019

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

In this paper, we present a high order finite difference solver for anisotropic diffusion problems based on the first-order hyperbolic system method. In particular, we demonstrate that the construction of a uniformly accurate fifth-order scheme that is independent of the degree of anisotropy is made straightforward by the hyperbolic method with an optimal length scale. We demonstrate that the gradients are computed simultaneously to the same order of accuracy as that of the solution variable by using weight compact finite difference schemes. Furthermore, the approach is extended to improve further the simulation of the magnetized electrons test case previously discussed in Refs. [J. Comput. Phys., 284 (2015) 59-69 and 374 (2018) 1120-1151]. Numerical results indicate that these schemes are capable of delivering high accuracy and the proposed approach is expected to allow the hyperbolic method to be successfully applied to a wide variety of linear and nonlinear problems with anisotropic diffusion.

show abstract

A first-order hyperbolic system approach for dispersion

Cited by 29 publications

References 27 publications

High-order gradients with the shifted boundary method: An embedded enriched mixed formulation for elliptic PDEs

High-order gradients with the shifted boundary method: An embedded enriched mixed formulation for elliptic PDEs

Hyperbolic model for free surface shallow water flows with effects of dispersion, vorticity and topography

First order hyperbolic approach for Anisotropic Diffusion equation

Contact Info

Product

Resources

About