Implicit-explicit methods for a class of nonlinear nonlocal gradient flow equations modelling collective behaviour

Bürger, Raimund; Inzunza, Daniel; Mulet, Pep; Villada, Luis Miguel

doi:10.1016/j.apnum.2019.04.018

Cited by 7 publications

(9 citation statements)

References 23 publications

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“…For instance, strong stability preserving (SSP) explicit RK schemes are a popular class of time integrators associated with a favorable stability constraint on the time step ∆t [25][26][27][28][29][33][34][35]. Alternatively, once could employ implicit-explicit Runge-Kutta (IMEX-RK) methods (see [33][34][35]), for which only the diffusion term is treated implicitly. In this case the stability condition on ∆t is less restrictive than for explicit discretizations, but a large system of nonlinear algebraic equations must be solved in each time step.…”

Section: Methodsmentioning

confidence: 99%

“…However, IMEX methods are more general than semi-implicit methods and are based on interlacing evaluations of the stages of particular explicit and implicit Runge-Kutta (RK) schemes [21,22]. We refer to [23,24] as the earliest references to IMEX methods in the context of time-dependent partial differential equations and [25][26][27][28][29][30][31][32][33][34][35] for various applications of IMEX-RK schemes.…”

Section: Introductionmentioning

confidence: 99%

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Exploring a Convection–Diffusion–Reaction Model of the Propagation of Forest Fires: Computation of Risk Maps for Heterogeneous Environments

et al. 2020

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The propagation of a forest fire can be described by a convection–diffusion–reaction problem in two spatial dimensions, where the unknowns are the local temperature and the portion of fuel consumed as functions of spatial position and time. This model can be solved numerically in an efficient way by a linearly implicit–explicit (IMEX) method to discretize the convection and nonlinear diffusion terms combined with a Strang-type operator splitting to handle the reaction term. This method is applied to several variants of the model with variable, nonlinear diffusion functions, where it turns out that increasing diffusivity (with respect to a given base case) significantly enlarges the portion of fuel burnt within a given time while choosing an equivalent constant diffusivity or a degenerate one produces comparable results for that quantity. In addition, the effect of spatial heterogeneity as described by a variable topography is studied. The variability of topography influences the local velocity and direction of wind. It is demonstrated how this variability affects the direction and speed of propagation of the wildfire and the location and size of the area of fuel consumed. The possibility to solve the base model efficiently is utilized for the computation of so-called risk maps. Here the risk associated with a given position in a sub-area of the computational domain is quantified by the rapidity of consumption of a given amount of fuel by a fire starting in that position. As a result, we obtain that, in comparison with the planar case and under the same wind conditions, the model predicts a higher risk for those areas where both the variability of topography (as expressed by the gradient of its height function) and the wind velocity are influential. In general, numerical simulations show that in all cases the risk map with for a non-planar topography includes areas with a reduced risk as well as such with an enhanced risk as compared to the planar case.

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Exploring a Convection–Diffusion–Reaction Model of the Propagation of Forest Fires: Computation of Risk Maps for Heterogeneous Environments

et al. 2020

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“…Explicit numerical schemes for (1) on a uniform Cartesian grid of meshwidth ∆x and time step ∆t are easy to implement but are associated with a Courant-Friedrichs-Lewy (CFL) stability condition [2] that imposes the proportionality ∆t ≈ ∆x 2 [3][4][5][6][7][8][9][10][11][12][13]. This restriction makes long-term simulations of ( 1)-( 3) on a uniform grid unacceptably slow.…”

Section: Scopementioning

confidence: 99%

“…Strong Stability Preserving (SSP) explicit Runge-Kutta methods are a popular class of time integrators whose use leads to a stronger stability constraint on ∆t, see [3][4][5][6][9][10][11]13]. An alternative to explicit RK schemes are implicit-explicit Runge-Kutta (IMEX-RK) methods (see [9][10][11]), for which only the diffusion term is treated implicitly. In this case the stability condition on ∆t is less severe than for explicit RK schemes, but a large number of nonlinear systems need to be solved.…”

Section: Notation and Semi-discrete Formulationmentioning

confidence: 99%

Implicit-Explicit Methods for a Convection-Diffusion-Reaction Model of the Propagation of Forest Fires

et al. 2020

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Numerical techniques for approximate solution of a system of reaction-diffusion-convection partial differential equations modeling the evolution of temperature and fuel density in a wildfire are proposed. These schemes combine linearly implicit-explicit Runge–Kutta (IMEX-RK) methods and Strang-type splitting technique to adequately handle the non-linear parabolic term and the stiffness in the reactive part. Weighted essentially non-oscillatory (WENO) reconstructions are applied to the discretization of the nonlinear convection term. Examples are focused on the applicative problem of determining the width of a firebreak to prevent the propagation of forest fires. Results illustrate that the model and numerical scheme provide an effective tool for defining that width and the parameters for control strategies of wildland fires.

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“…Several works have devoted to the efficient numerical methods for the treatment of nonlinear problems in recent years. For example, we can refer to the Euler implicit–explicit methods (Bürger et al , 2019; Giacomo and Lorenzo, 2013; He, 2013), the Newton scheme (Durango and Novo, 2018; Liu et al , 2012; Oliver et al , 2011) and the Oseen method (An, 2014; Shang, 2018). The main motivation of designing these numerical schemes is to provide a rapid and efficient mean of solving the considered problem.…”

Section: Introductionmentioning

confidence: 99%

Stability and convergence of iterative finite element methods for the thermally coupled incompressible MHD flow

Yang

Zhang

2020

HFF

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Purpose The purpose of this paper is to propose three iterative finite element methods for equations of thermally coupled incompressible magneto-hydrodynamics (MHD) on 2D/3D bounded domain. The detailed theoretical analysis and some numerical results are presented. The main results show that the Stokes iterative method has the strictest restrictions on the physical parameters, and the Newton’s iterative method has the higher accuracy and the Oseen iterative method is stable unconditionally. Design/methodology/approach Three iterative finite element methods have been designed for the thermally coupled incompressible MHD flow on 2D/3D bounded domain. The Oseen iterative scheme includes solving a linearized steady MHD and Oseen equations; unconditional stability and optimal error estimates of numerical approximations at each iterative step are established under the uniqueness condition. Stability and convergence of numerical solutions in Newton and Stokes’ iterative schemes are also analyzed under some strong uniqueness conditions. Findings This work was supported by the NSF of China (No. 11971152). Originality/value This paper presents the best choice for solving the steady thermally coupled MHD equations with different physical parameters.

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Implicit-explicit methods for a class of nonlinear nonlocal gradient flow equations modelling collective behaviour

Cited by 7 publications

References 23 publications

Exploring a Convection–Diffusion–Reaction Model of the Propagation of Forest Fires: Computation of Risk Maps for Heterogeneous Environments

Exploring a Convection–Diffusion–Reaction Model of the Propagation of Forest Fires: Computation of Risk Maps for Heterogeneous Environments

Implicit-Explicit Methods for a Convection-Diffusion-Reaction Model of the Propagation of Forest Fires

Stability and convergence of iterative finite element methods for the thermally coupled incompressible MHD flow

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