High Order Semi-implicit Schemes for Time Dependent Partial Differential Equations

Boscarino, Sebastiano; Filbet, Francis; Russo, Giovanni

doi:10.1007/s10915-016-0168-y

Cited by 117 publications

(161 citation statements)

References 39 publications

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“…High order in time can be obtained by adopting IMEX-RK schemes withb = b. For a more general description of the relation between IMEX and ARK methods, see [4].…”

Section: Linearly Implicit Imex-rk Methodsmentioning

confidence: 99%

“…It is easy to show that ARK schemes can been seen as a particular case of IMEX-RK schemes, obtained by introducing two families of stage values, respectively, for the explicit and implicit parts (see [4]). ARK schemes allow a linearly implicit form which is particularly efficient, since the implicit term is linear and entails no loss of order of accuracy or stability.…”

Section: Time Integratormentioning

confidence: 99%

“…The two schemes H-LDIRK2(2,2,2) and SSP-LDIRK (3,3,2) were introduced in the context of hyperbolic systems with stiff relaxation [25]. Table 1 The second-order IMEX-RK schemes we adopt in this paper [4,25]. Downloaded 07/18/15 to 130.132.123.28.…”

Section: Some Second-order Imex-rk Schemesmentioning

confidence: 99%

See 2 more Smart Citations

Linearly Implicit IMEX Runge--Kutta Methods for a Class of Degenerate Convection-Diffusion Problems

Boscarino¹,

Bürger²,

Mulet³

et al. 2015

SIAM J. Sci. Comput.

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Multispecies kinematic flow models with strongly degenerate diffusive corrections give rise to systems of nonlinear convection-diffusion equations of arbitrary size. Applications of these systems include models of polydisperse sedimentation and multiclass traffic flow. Implicit-explicit (IMEX) Runge-Kutta (RK) methods are suitable for the solution of these convection-diffusion problems since the stability restrictions, coming from the explicitly treated convective part, are much less severe than those that would be deduced from an explicit treatment of the diffusive term. These schemes usually combine an explicit RK scheme for the time integration of the convective part with a diagonally implicit one for the diffusive part. In [R. Bürger, P. Mulet, and L. M. Villada, SIAM J. Sci. Comput., 35 (2013), pp. B751-B777] a scheme of this type is proposed, where the nonlinear and nonsmooth systems of algebraic equations arising in the implicit treatment of the degenerate diffusive part are solved by smoothing of the diffusion coefficients combined with a Newton-Raphson method with line search. This nonlinearly implicit method is robust but associated with considerable effort of implementation and possibly CPU time. To overcome these shortcomings while keeping the advantageous stability properties of IMEX-RK methods, a second variant of these methods is proposed in which the diffusion terms are discretized in a way that more carefully distinguishes between stiff and nonstiff dependence, such that in each time step only a linear system needs to be solved still maintaining high order accuracy in time, which makes these methods much simpler to implement. In a series of examples of polydisperse sedimentation and multiclass traffic flow, it is demonstrated that these new linearly implicit IMEX-RK schemes approximate the same solutions as the nonlinearly implicit versions, and in many cases these schemes are more efficient.

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“…High order in time can be obtained by adopting IMEX-RK schemes withb = b. For a more general description of the relation between IMEX and ARK methods, see [4].…”

Section: Linearly Implicit Imex-rk Methodsmentioning

confidence: 99%

Section: Time Integratormentioning

confidence: 99%

See 1 more Smart Citation

Linearly Implicit IMEX Runge--Kutta Methods for a Class of Degenerate Convection-Diffusion Problems

Boscarino¹,

Bürger²,

Mulet³

et al. 2015

SIAM J. Sci. Comput.

Self Cite

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“…The approach has also been studied more generally in more recent work [4]. For suspension flow, a semi-implicit method was used to simulate a coupled system with simpler flux functions and studied numerically [10,16].…”

Section: Numerical Schemementioning

confidence: 99%

Surface tension effects for particle settling and resuspension in viscous thin films

et al. 2018

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We consider flow of a thin film on an incline with negatively buoyant particles. We derive a onedimensional lubrication model, including the effect of surface tension, which is a nontrivial extension of a previous model (Murisic et. al [J. Fluid Mech. 2013]). We show that the surface tension, in the form of high order derivatives, not only regularizes the previous model as a high order diffusion, but also modifies the fluxes. As a result, it leads to a different stratification in the particle concentration along the direction perpendicular to the motion of the fluid mixture. The resulting equations are of mixed hyperbolic-parabolic type and different from the well-known lubrication theory for a clear fluid or fluid with surfactant. To study the system numerically, we formulate a semi-implicit scheme that is able to preserve the particle maximum packing fraction. We show extensive numerical results for this model including a qualitative comparison with two-dimensional laboratory experiments.

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“…In fact, after the pioneering work of Crouzeix , numerical integrators that deal implicitly with

D (bold-italicu)

and explicitly with

C (bold-italicu)

can be used with a time step restriction dictated by the convective term alone. These schemes, apart from having been profusely used in convection–diffusion problems and convection problems with stiff reaction term , have been used recently to deal with stiff terms in hyperbolic systems with relaxation . Finally, we mention that many authors have proposed IMEX‐RK scheme for the solution of semi‐discretized PDEs .…”

Section: Introductionmentioning

confidence: 99%

Implicit–explicit schemes for nonlinear nonlocal equations with a gradient flow structure in one space dimension

Bürger

Inzunza

Mulet

et al. 2019

Numerical Methods Partial

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Nonlinear convection–diffusion equations with nonlocal flux and possibly degenerate diffusion arise in various contexts including interacting gases, porous media flows, and collective behavior in biology. Their numerical solution by an explicit finite difference method is costly due to the necessity of discretizing a local spatial convolution for each evaluation of the convective numerical flux, and due to the disadvantageous Courant–Friedrichs–Lewy (CFL) condition incurred by the diffusion term. Based on explicit schemes for such models devised in the study of Carrillo et al. a second‐order implicit–explicit Runge–Kutta (IMEX‐RK) method can be formulated. This method avoids the restrictive time step limitation of explicit schemes since the diffusion term is handled implicitly, but entails the necessity to solve nonlinear algebraic systems in every time step. It is proven that this method is well defined. Numerical experiments illustrate that for fine discretizations it is more efficient in terms of reduction of error versus central processing unit time than the original explicit method. One of the test cases is given by a strongly degenerate parabolic, nonlocal equation modeling aggregation in study of Betancourt et al. This model can be transformed to a local partial differential equation that can be solved numerically easily to generate a reference solution for the IMEX‐RK method, but is limited to one space dimension.

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High Order Semi-implicit Schemes for Time Dependent Partial Differential Equations

Cited by 117 publications

References 39 publications

Linearly Implicit IMEX Runge--Kutta Methods for a Class of Degenerate Convection-Diffusion Problems

Linearly Implicit IMEX Runge--Kutta Methods for a Class of Degenerate Convection-Diffusion Problems

Surface tension effects for particle settling and resuspension in viscous thin films

Implicit–explicit schemes for nonlinear nonlocal equations with a gradient flow structure in one space dimension

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