Third order positivity-preserving direct discontinuous Galerkin method with interface correction for chemotaxis Keller-Segel equations

Qiu, Changxin; Li, Qingyuan; Yan, Jue

doi:10.1016/j.jcp.2021.110191

Cited by 13 publications

(9 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Even though the scheme (23) for ρ does not involve any auxiliary variable g, the division by M n i is still needed in (23). Moreover, (22) gives a symmetric positive definite linear system but (23) does not. In practice, both can be solved by preconditioned conjugate gradient methods with efficient inversion of Laplacian as a preconditioner, see Section 7 in [17] for implementation details.…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…In the case of the Keller-Segel system, in addition to (22) (the discretization for ( 7)) one also needs to discretize the equation ( 6). Here we consider α > 0 and the homogeneous Neumann boundary condition ∇c • n| ∂Ω = 0.…”

Section: The Full Scheme For the Keller-segel Systemmentioning

confidence: 99%

“…Nonetheless, for the equations we are interested in here, i.e., an operator like −∇ • (a(x)∇u) with a scalar coefficient a(x), since there are no mixed second order derivatives involved, the same proof in [17] applies to show that the fourth order accuracy also holds for Neumann boundary conditions of elliptic equations, see [14] for the parabolic case. So for both (22) and ( 24), we will refer to the Q 2 scheme as the fourth order accurate spatial discretization, i.e., it is a fourth order accurate scheme for solving a steady state problem.…”

Section: Accuracy Of the Spatial Discretizationmentioning

confidence: 99%

“…The first kind of methods are fully explicit schemes. For a scalar convection-diffusion equation such as (5), there are quite a few explicit positivity-preserving schemes [15,22,24,28], however with a small time step constraint ∆t = O(∆x 2 ) which is unacceptable in applications requiring long time simulation. Most importantly, it is usually quite difficult to establish energy dissipation in these positivity-preserving schemes.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Positivity-preserving and energy-dissipative finite difference schemes for the Fokker-Planck and Keller-Segel equations

Hu¹,

Zhang²

2021

Preprint

View full text Add to dashboard Cite

In this work, we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the linear Fokker-Planck equation and the Keller-Segel equations. The two proposed schemes are first order accurate in time, explicitly solvable, and second order and fourth order accurate in space, which are obtained via finite difference implementation of the classical continuous finite element method. The fully discrete schemes are proven positivity-preserving and energy-dissipative: the second order scheme can achieve so unconditionally; the fourth order scheme only requires a mild time step and mesh size constraint. Furthermore, the fourth order scheme is the first high order spatial discretization that can achieve both positivity and energy decay properties, which is suitable for long time simulation and to obtain accurate steady state solutions.

show abstract

Section: Spatial Discretizationmentioning

confidence: 99%

Section: The Full Scheme For the Keller-segel Systemmentioning

confidence: 99%

Section: Accuracy Of the Spatial Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Positivity-preserving and energy-dissipative finite difference schemes for the Fokker-Planck and Keller-Segel equations

Hu¹,

Zhang²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…To construct bound-preserving schemes for equation (1.1), we can first consider bound-preserving schemes for a convection-diffusion equation, e.g., F (φ) ≡ 0. In the literature, there are many fully explicit high order accurate bound-preserving schemes for a scalar convection-diffusion equation [3,7,9,17,19,20,22]. In these schemes, the time discretizations are high order explicit time strong stability preserving (SSP) Runge-Kutta and multistep methods, which are convex combinations of forward Euler steps.…”

mentioning

confidence: 99%

Discrete Maximum principle of a high order finite difference scheme for a generalized Allen-Cahn equation

Shen¹,

Zhang²

2021

Preprint

View full text Add to dashboard Cite

We consider solving a generalized Allen-Cahn equation coupled with a passive convection for a given incompressible velocity field. The numerical scheme consists of the first order accurate stabilized implicit explicit time discretization and a fourth order accurate finite difference scheme, which is obtained from the finite difference formulation of the Q 2 spectral element method. We prove that the discrete maximum principle holds under suitable mesh size and time step constraints. The same result also applies to construct a bound-preserving scheme for any passive convection with an incompressible velocity field.

show abstract

A Generalized Framework for Direct Discontinuous Galerkin Methods for Nonlinear Diffusion Equations

Danis

Yan

2023

J Sci Comput

View full text Add to dashboard Cite

Third order positivity-preserving direct discontinuous Galerkin method with interface correction for chemotaxis Keller-Segel equations

Cited by 13 publications

References 42 publications

Positivity-preserving and energy-dissipative finite difference schemes for the Fokker-Planck and Keller-Segel equations

Positivity-preserving and energy-dissipative finite difference schemes for the Fokker-Planck and Keller-Segel equations

Discrete Maximum principle of a high order finite difference scheme for a generalized Allen-Cahn equation

A Generalized Framework for Direct Discontinuous Galerkin Methods for Nonlinear Diffusion Equations

Contact Info

Product

Resources

About