Issues

Journal & Issues

Volume 13 (2022): Issue 1 (January 2022)

Volume 12 (2021): Issue 1 (January 2021)

Volume 11 (2020): Issue 1 (January 2020)

Volume 10 (2019): Issue 2 (January 2019)
Special Issue on Mathematical Models and Methods in Biology, Medicine and Physiology. Guest Editors: Michele Piana, Luigi Preziosi

Volume 10 (2019): Issue 1 (February 2019)

Volume 9 (2018): Issue 2 (December 2018)
Special Issue on Mathematical modelling for complex systems: multi-agents methods. Guest Editor: Elena De Angelis

Volume 9 (2018): Issue 1 (February 2018)

Volume 8 (2017): Issue 1 (March 2017)

Volume 7 (2016): Issue 3 (September 2016)
"Special Issue on New Trends in Semi-Lagrangian Methods, Guest Editors: Luca Bonaventura, Maurizio Falcone and Roberto Ferretti

Volume 7 (2016): Issue 2 (June 2016)
Special Issue on Constitutive Equations for Heat Conduction in Nanosystems and Non-equilibrium Processes. Guest Editors: Vito Antonio Cimmelli and David Jou

Volume 7 (2016): Issue 1 (January 2016)
Open Issue

Journal Details
Format
Journal
eISSN
2038-0909
First Published
15 Dec 2014
Publication timeframe
1 time per year
Languages
English

Search

Volume 7 (2016): Issue 3 (September 2016)
"Special Issue on New Trends in Semi-Lagrangian Methods, Guest Editors: Luca Bonaventura, Maurizio Falcone and Roberto Ferretti

Journal Details
Format
Journal
eISSN
2038-0909
First Published
15 Dec 2014
Publication timeframe
1 time per year
Languages
English

Search

8 Articles
access type Open Access

Introduction

Published Online: 01 Oct 2016
Page range: 1 - 3

Abstract

access type Open Access

Semi-implicit semi-Lagrangian modelling of the atmosphere: a Met Office perspective

Published Online: 01 Oct 2016
Page range: 4 - 25

Abstract

Abstract

The semi-Lagrangian numerical method, in conjunction with semi-implicit time integration, provides numerical weather prediction models with numerical stability for large time steps, accurate modes of interest, and good representation of hydrostatic and geostrophic balance. Drawing on the legacy of dynamical cores at the Met Office, the use of the semi-implicit semi-Lagrangian method in an operational numerical weather prediction context is surveyed, together with details of the solution approach and associated issues and challenges. The numerical properties and performance of the current operational version of the Met Office’s numerical model are then investigated in a simplified setting along with the impact of different modelling choices.

Keywords

  • Semi-Lagrangian method
  • semi-implicit method
  • dynamical core
  • normal mode analysis
access type Open Access

Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

Published Online: 01 Oct 2016
Page range: 26 - 55

Abstract

Abstract

We review in this paper the development of Lagrange-Galerkin (LG) methods to integrate the incompressible Navier-Stokes equations (NSEs) for engineering applications. These methods were introduced in the computational fluid dynamics community in the early eighties of the past century, and at that time they were considered good methods for both their theoretical stability properties and the way of dealing with the nonlinear terms of the equations; however, the numerical experience gained with the application of LG methods to different problems has identified drawbacks of them, such as the calculation of specific integrals that arise in their formulation and the calculation of the ow trajectories, which somehow have hampered the applicability of LG methods. In this paper, we focus on these issues and summarize the convergence results of LG methods; furthermore, we shall briefly introduce a new stabilized LG method suitable for high Reynolds numbers.

Keywords

  • Lagrange-Galerkin
  • finite elements
  • Navier-Stokes
access type Open Access

Flux form Semi-Lagrangian methods for parabolic problems

Published Online: 01 Oct 2016
Page range: 56 - 73

Abstract

Abstract

A semi-Lagrangian method for parabolic problems is proposed, that extends previous work by the authors to achieve a fully conservative, flux-form discretization of linear and nonlinear diffusion equations. A basic consistency and stability analysis is proposed. Numerical examples validate the proposed method and display its potential for consistent semi-Lagrangian discretization of advection diffusion and nonlinear parabolic problems.

Keywords

  • Semi-Lagrangian methods
  • Flux-Form Semi-Lagrangian methods
  • Diffusion equations
  • Divergence form
access type Open Access

Two conservative multi-tracer efficient semi-Lagrangian schemes for multiple processor systems integrated in a spectral element (climate) dynamical core

Published Online: 01 Oct 2016
Page range: 74 - 98

Abstract

Abstract

In today’s atmospheric numerical modeling, scalable and highly accurate numerical schemes are of particular interest. To address these issues Galerkin schemes, such as the spectral element method, have received more attention in the last decade. They also provide other state-of-the-art capabilities such as improved conservation. However, the tracer transport of hundreds of tracers, e.g., in the chemistry version of the Community Atmosphere Model, is still a performance bottleneck. Therefore, we consider two conservative semi-Lagrangian schemes. Both are designed to be multi-tracer efficient, third order accurate, and allow significantly longer time steps than explicit Eulerian formulations. We address the difficulties arising on the cubed-sphere projection and on parallel computers and show the high scalability of our approach. Additionally, we use the two schemes for the transport of passive tracers in a dynamical core and compare our results with a current spectral element tracer transport advection used by the High-Order Method Modeling Environment.

Keywords

  • transport scheme
  • spherical geometry
  • cubed-sphere grid
  • conservative semi-Lagrangian
  • spectral element method
  • error
  • parallel scalability
  • performance
access type Open Access

The semi-Lagrangian method on curvilinear grids

Published Online: 01 Oct 2016
Page range: 99 - 137

Abstract

Abstract

We study the semi-Lagrangian method on curvilinear grids. The classical backward semi-Lagrangian method [1] preserves constant states but is not mass conservative. Natural reconstruction of the field permits nevertheless to have at least first order in time conservation of mass, even if the spatial error is large. Interpolation is performed with classical cubic splines and also cubic Hermite interpolation with arbitrary reconstruction order of the derivatives. High odd order reconstruction of the derivatives is shown to be a good ersatz of cubic splines which do not behave very well as time step tends to zero. A conservative semi-Lagrangian scheme along the lines of [2] is then described; here conservation of mass is automatically satisfied and constant states are shown to be preserved up to first order in time.

Keywords

  • Vlasov equation
  • guiding center model
  • semi-Lagrangian method
  • curvilinear grid
  • mapped grid
access type Open Access

Boundary conditions for semi-Lagrangian methods for the BGK model

Published Online: 01 Oct 2016
Page range: 138 - 164

Abstract

Abstract

A new class of high-order accuracy numerical methods based on a semi-Lagrangian formulation for the BGK model of the Boltzmann equation has been recently proposed in [1]. In this paper semi-Lagrangian schemes for the BGK equation have been extended to treat boundary conditions, in particular the diffusive ones. Two different techniques are proposed, using or avoiding iterative procedures. Numerical simulations illustrate the accuracy properties of these approaches and the agreement with the results available in literature.

Keywords

  • BGK equation
  • semi-Lagrangian methods
  • high-order schemes
  • boundary conditions
access type Open Access

A mass conservative TR-BDF2 semi-implicit semi-Lagrangian DG discretization of the shallow water equations on general structured meshes of quadrilaterals

Published Online: 01 Oct 2016
Page range: 165 - 190

Abstract

Abstract

As an extension of a previous work considering a fully advective formulation on Cartesian meshes, a mass conservative discretization approach is presented here for the shallow water equations, based on discontinuous finite elements on general structured meshes of quadrilaterals. A semi-implicit time integration is performed by employing the TR-BDF2 scheme and is combined with the semi-Lagrangian technique for the momentum equation only. Indeed, in order to simplify the derivation of the discrete linear Helmoltz equation to be solved at each time-step, a non-conservative formulation of the momentum equation is employed. The Eulerian flux form is considered instead for the continuity equation in order to ensure mass conservation. Numerical results show that on distorted meshes and for relatively high polynomial degrees, the proposed numerical method fully conserves mass and presents a higher level of accuracy than a standard off-centered Crank Nicolson approach. This is achieved without any significant imprinting of the mesh distortion on the solution.

Keywords

  • discontinuous Galerkin methods
  • semi-implicit discretizations
  • semi-Lagrangian discretizations
  • shallow water equations
  • qudrilateral meshes
8 Articles
access type Open Access

Introduction

Published Online: 01 Oct 2016
Page range: 1 - 3

Abstract

access type Open Access

Semi-implicit semi-Lagrangian modelling of the atmosphere: a Met Office perspective

Published Online: 01 Oct 2016
Page range: 4 - 25

Abstract

Abstract

The semi-Lagrangian numerical method, in conjunction with semi-implicit time integration, provides numerical weather prediction models with numerical stability for large time steps, accurate modes of interest, and good representation of hydrostatic and geostrophic balance. Drawing on the legacy of dynamical cores at the Met Office, the use of the semi-implicit semi-Lagrangian method in an operational numerical weather prediction context is surveyed, together with details of the solution approach and associated issues and challenges. The numerical properties and performance of the current operational version of the Met Office’s numerical model are then investigated in a simplified setting along with the impact of different modelling choices.

Keywords

  • Semi-Lagrangian method
  • semi-implicit method
  • dynamical core
  • normal mode analysis
access type Open Access

Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

Published Online: 01 Oct 2016
Page range: 26 - 55

Abstract

Abstract

We review in this paper the development of Lagrange-Galerkin (LG) methods to integrate the incompressible Navier-Stokes equations (NSEs) for engineering applications. These methods were introduced in the computational fluid dynamics community in the early eighties of the past century, and at that time they were considered good methods for both their theoretical stability properties and the way of dealing with the nonlinear terms of the equations; however, the numerical experience gained with the application of LG methods to different problems has identified drawbacks of them, such as the calculation of specific integrals that arise in their formulation and the calculation of the ow trajectories, which somehow have hampered the applicability of LG methods. In this paper, we focus on these issues and summarize the convergence results of LG methods; furthermore, we shall briefly introduce a new stabilized LG method suitable for high Reynolds numbers.

Keywords

  • Lagrange-Galerkin
  • finite elements
  • Navier-Stokes
access type Open Access

Flux form Semi-Lagrangian methods for parabolic problems

Published Online: 01 Oct 2016
Page range: 56 - 73

Abstract

Abstract

A semi-Lagrangian method for parabolic problems is proposed, that extends previous work by the authors to achieve a fully conservative, flux-form discretization of linear and nonlinear diffusion equations. A basic consistency and stability analysis is proposed. Numerical examples validate the proposed method and display its potential for consistent semi-Lagrangian discretization of advection diffusion and nonlinear parabolic problems.

Keywords

  • Semi-Lagrangian methods
  • Flux-Form Semi-Lagrangian methods
  • Diffusion equations
  • Divergence form
access type Open Access

Two conservative multi-tracer efficient semi-Lagrangian schemes for multiple processor systems integrated in a spectral element (climate) dynamical core

Published Online: 01 Oct 2016
Page range: 74 - 98

Abstract

Abstract

In today’s atmospheric numerical modeling, scalable and highly accurate numerical schemes are of particular interest. To address these issues Galerkin schemes, such as the spectral element method, have received more attention in the last decade. They also provide other state-of-the-art capabilities such as improved conservation. However, the tracer transport of hundreds of tracers, e.g., in the chemistry version of the Community Atmosphere Model, is still a performance bottleneck. Therefore, we consider two conservative semi-Lagrangian schemes. Both are designed to be multi-tracer efficient, third order accurate, and allow significantly longer time steps than explicit Eulerian formulations. We address the difficulties arising on the cubed-sphere projection and on parallel computers and show the high scalability of our approach. Additionally, we use the two schemes for the transport of passive tracers in a dynamical core and compare our results with a current spectral element tracer transport advection used by the High-Order Method Modeling Environment.

Keywords

  • transport scheme
  • spherical geometry
  • cubed-sphere grid
  • conservative semi-Lagrangian
  • spectral element method
  • error
  • parallel scalability
  • performance
access type Open Access

The semi-Lagrangian method on curvilinear grids

Published Online: 01 Oct 2016
Page range: 99 - 137

Abstract

Abstract

We study the semi-Lagrangian method on curvilinear grids. The classical backward semi-Lagrangian method [1] preserves constant states but is not mass conservative. Natural reconstruction of the field permits nevertheless to have at least first order in time conservation of mass, even if the spatial error is large. Interpolation is performed with classical cubic splines and also cubic Hermite interpolation with arbitrary reconstruction order of the derivatives. High odd order reconstruction of the derivatives is shown to be a good ersatz of cubic splines which do not behave very well as time step tends to zero. A conservative semi-Lagrangian scheme along the lines of [2] is then described; here conservation of mass is automatically satisfied and constant states are shown to be preserved up to first order in time.

Keywords

  • Vlasov equation
  • guiding center model
  • semi-Lagrangian method
  • curvilinear grid
  • mapped grid
access type Open Access

Boundary conditions for semi-Lagrangian methods for the BGK model

Published Online: 01 Oct 2016
Page range: 138 - 164

Abstract

Abstract

A new class of high-order accuracy numerical methods based on a semi-Lagrangian formulation for the BGK model of the Boltzmann equation has been recently proposed in [1]. In this paper semi-Lagrangian schemes for the BGK equation have been extended to treat boundary conditions, in particular the diffusive ones. Two different techniques are proposed, using or avoiding iterative procedures. Numerical simulations illustrate the accuracy properties of these approaches and the agreement with the results available in literature.

Keywords

  • BGK equation
  • semi-Lagrangian methods
  • high-order schemes
  • boundary conditions
access type Open Access

A mass conservative TR-BDF2 semi-implicit semi-Lagrangian DG discretization of the shallow water equations on general structured meshes of quadrilaterals

Published Online: 01 Oct 2016
Page range: 165 - 190

Abstract

Abstract

As an extension of a previous work considering a fully advective formulation on Cartesian meshes, a mass conservative discretization approach is presented here for the shallow water equations, based on discontinuous finite elements on general structured meshes of quadrilaterals. A semi-implicit time integration is performed by employing the TR-BDF2 scheme and is combined with the semi-Lagrangian technique for the momentum equation only. Indeed, in order to simplify the derivation of the discrete linear Helmoltz equation to be solved at each time-step, a non-conservative formulation of the momentum equation is employed. The Eulerian flux form is considered instead for the continuity equation in order to ensure mass conservation. Numerical results show that on distorted meshes and for relatively high polynomial degrees, the proposed numerical method fully conserves mass and presents a higher level of accuracy than a standard off-centered Crank Nicolson approach. This is achieved without any significant imprinting of the mesh distortion on the solution.

Keywords

  • discontinuous Galerkin methods
  • semi-implicit discretizations
  • semi-Lagrangian discretizations
  • shallow water equations
  • qudrilateral meshes

Plan your remote conference with Sciendo