A generic framework for time-stepping partial differential equations (PDEs): general linear methods, object-oriented implementation and application to fluid problems

Vos, Peter; Eskilsson, Claes; Bolis, A.; Chun, Seok-Ju; Kirby, Robert M.; Sherwin, Spencer J.

doi:10.1080/10618562.2011.575368

Cited by 46 publications

(31 citation statements)

References 24 publications

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“…This includes core functionality, such as IO, time-stepping [20] and common initialisation routines, useful in quickly constructing a solver using the Nektar++ framework. It contains a library of applicationindependent modules for implementing diffusion and advection terms as well as a number of Driver modules which implement general high-level algorithms, such as an Arnoldi method for performing various stability analyses [21].…”

Section: Solverutils Librarymentioning

confidence: 99%

“…Given the complexity and highly nonlinear form of these equations, we adopt a fully explicit formulation to discretise the equations in time, allowing us to use any of the explicit timestepping algorithms implemented through the general linear methods framework [20], including 2nd and 4th order Runge-Kutta methods. Finally, in order to stabilise the flow in the presence of discontinuities we utilise a shock capturing technique which makes use of artificial viscosity to damp oscillations in the solution, in conjunction with a discontinuity sensor adapted from the approach taken in [30] to decide where the addition of artificial viscosity is needed.…”

Section: External Aerodynamicsmentioning

confidence: 99%

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Nektar++: An open-source spectral/hp element framework

Cantwell

Moxey

Comerford

et al. 2015

Computer Physics Communications

500

343

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a b s t r a c tNektar++ is an open-source software framework designed to support the development of highperformance scalable solvers for partial differential equations using the spectral/hp element method. High-order methods are gaining prominence in several engineering and biomedical applications due to their improved accuracy over low-order techniques at reduced computational cost for a given number of degrees of freedom. However, their proliferation is often limited by their complexity, which makes these methods challenging to implement and use. Nektar++ is an initiative to overcome this limitation by encapsulating the mathematical complexities of the underlying method within an efficient C++ framework, making the techniques more accessible to the broader scientific and industrial communities. The software supports a variety of discretisation techniques and implementation strategies, supporting methods research as well as application-focused computation, and the multi-layered structure of the framework allows the user to embrace as much or as little of the complexity as they need. The libraries capture the mathematical constructs of spectral/hp element methods, while the associated collection of pre-written PDE solvers provides out-of-the-box application-level functionality and a template for users who wish to develop solutions for addressing questions in their own scientific domains. Program summaryProgram title: Nektar++ Catalogue identifier: AEVV_v1_0Program summary URL:

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Section: Solverutils Librarymentioning

confidence: 99%

Section: External Aerodynamicsmentioning

confidence: 99%

Nektar++: An open-source spectral/hp element framework

Cantwell

Moxey

Comerford

et al. 2015

Computer Physics Communications

500

343

View full text Add to dashboard Cite

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“…For unsteady problems the equations can be time-marched using one of a number of implicit or implicit–explicit (IMEX) time integration schemes [28], implemented using general linear methods [29]. In summary, the PDE is arranged in the form

\frac{\partial u}{\partial t} = f false(u false) + g false(u false),

where

f false(u false)

is typically nonlinear and therefore should be evaluated explicitly, and

g false(u false)

is stiff and therefore best evaluated implicitly so as to avoid excessively small time-steps.…”

Section: Formulationmentioning

confidence: 99%

High-order spectral/hp element discretisation for reaction–diffusion problems on surfaces: Application to cardiac electrophysiology

Cantwell

Yakovlev

Kirby

et al. 2014

Journal of Computational Physics

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We present a numerical discretisation of an embedded two-dimensional manifold using high-order continuous Galerkin spectral/hp elements, which provide exponential convergence of the solution with increasing polynomial order, while retaining geometric flexibility in the representation of the domain. Our work is motivated by applications in cardiac electrophysiology where sharp gradients in the solution benefit from the high-order discretisation, while the computational cost of anatomically-realistic models can be significantly reduced through the surface representation and use of high-order methods. We describe and validate our discretisation and provide a demonstration of its application to modelling electrochemical propagation across a human left atrium.

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“…The main capabilities of the explicit and implicit compressible flow solvers are summarized here. The explicit solver can use forward Euler, Adams Bashforth [40] and 2nd-4th order Runge-Kutta for temporal discretization, while the implicit solver can use backward Euler, 2nd order BDF (Backward differentiation formula) and 2nd-4th order SDIRK. The other aspects of the solvers are summarized as follows:…”

Section: Solver Capabilities and Code Verificationmentioning

confidence: 99%

Nektar++: Design and implementation of an implicit, spectral/hp element, compressible flow solver using a Jacobian-free Newton Krylov approach

Yan

Pan

Castiglioni

et al. 2021

Computers & Mathematics with Applications

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At high Reynolds numbers the use of explicit in time compressible flow simulations with spectral/hp element discretization can become significantly limited by time step. To alleviate this limitation we extend the capability of the spectral/hp element open-source software framework, Nektar++, to include an implicit discontinuous Galerkin compressible flow solver. The integration in time is carried out by a singly diagonally implicit Runge-Kutta method. The non-linear system arising from the implicit time integration is iteratively solved by the Jacobian-free Newton Krylov (JFNK) method. A favorable feature of the JFNK approach is its extensive use of the explicit operators available from the previous explicit in time implementation. The functionalities of different building blocks of the implicit solver are analyzed from the point of view of software design and placed in appropriate hierarchical levels in the C++ libraries. In the detailed implementation, the contributions of different parts of the solver to computational cost, memory consumption and programming complexity are also analyzed. A combination of analytical and numerical methods is adopted to simplify the programming complexity in forming the preconditioning matrix. The solver is verified and tested using cases such as manufactured compressible tex, turbulent flow over a circular cylinder at Re = 3900 and shock wave boundary-layer interaction. The results show that the implicit solver can speed-up the simulations while maintaining good simulation accuracy.

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A generic framework for time-stepping partial differential equations (PDEs): general linear methods, object-oriented implementation and application to fluid problems

Cited by 46 publications

References 24 publications

Nektar++: An open-source spectral/hp element framework

Nektar++: An open-source spectral/hp element framework

High-order spectral/hp element discretisation for reaction–diffusion problems on surfaces: Application to cardiac electrophysiology

Nektar++: Design and implementation of an implicit, spectral/hp element, compressible flow solver using a Jacobian-free Newton Krylov approach

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