# A new Lagrange multiplier approach for constructing structure-preserving schemes, II. bound preserving

@article{Cheng2021ANL, title={A new Lagrange multiplier approach for constructing structure-preserving schemes, II. bound preserving}, author={Qing Cheng and Jie Shen}, journal={ArXiv}, year={2021}, volume={abs/2109.12479} }

In the second part of this series, we use the Lagrange multiplier approach proposed in the first part [7] to construct efficient and accurate bound and/or mass preserving schemes for a class of semi-linear and quasi-linear parabolic equations. We establish stability results under a general setting, and carry out an error analysis for a secondorder bound preserving scheme with a hybrid spectral discretization in space. We apply our approach to several typical PDEs which preserve bound and/or… Expand

#### Figures and Tables from this paper

#### References

SHOWING 1-10 OF 32 REFERENCES

A new Lagrange multiplier approach for constructing structure preserving schemes, I. positivity preserving

- Mathematics, Computer Science
- 2021

A new Lagrange multiplier approach to construct positivity preserving schemes for parabolic type equations and is not restricted to any particular spatial discretization and can be combined with various time discretized schemes. Expand

Construction and Convergence Study of Schemes Preserving the Elliptic Local Maximum Principle

- Mathematics, Computer Science
- SIAM J. Numer. Anal.
- 2011

We present a method to approximate (in any space dimension) diffusion equations with schemes having a specific structure; this structure ensures that the discrete local maximum and minimum principles… Expand

Physical-bound-preserving finite volume methods for the Nagumo equation on distorted meshes

- Computer Science, Mathematics
- Comput. Math. Appl.
- 2019

It is proved that the numerical solution of the resulting scheme can preserve the bound of the solution for the Nagumo equation on distorted meshes. Expand

A bound-preserving high order scheme for variable density incompressible Navier-Stokes equations

- Computer Science, Physics
- J. Comput. Phys.
- 2021

This paper considers a combination of a continuous finite element method for momentum evolution and a bound-preserving DG method for density evolution, using fully explicit and explicit-implicit strong stability preserving Runge-Kutta methods. Expand

A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials

- Computer Science, Mathematics
- J. Comput. Phys.
- 2018

A high order discontinuous Galerkin method for a class of time-dependent second order partial differential equations governed by a decaying entropy, which covers important cases such as Fokker–Planck type equations and aggregation models. Expand

Maximum bound principles for a class of semilinear parabolic equations and exponential time differencing schemes

- Computer Science, Mathematics
- SIAM Rev.
- 2021

It is demonstrated that the abstract framework and analysis techniques developed here offer an effective and unified approach to study the maximum bound principle of the abstract evolution equation that cover a wide variety of well-known models and their numerical discretization schemes. Expand

Positivity Preserving Limiters for Time-Implicit Higher Order Accurate Discontinuous Galerkin Discretizations

- Mathematics, Computer Science
- SIAM J. Sci. Comput.
- 2019

An efficient active set semi-smooth Newton method is developed that is suitable for the KKT formulation of time-implicit positivity preserving DG discretization and is demonstrated for several nonlinear scalar conservation laws. Expand

Maximum principle and uniform convergence for the finite element method

- Mathematics
- 1973

Abstract The solution of the Dirichlet boundary value problem over a polyhedral domain Ω ⊂ R n , n ≥ 2, associated with a second-order elliptic operator, is approximated by the simplest finite… Expand

Maximum-Principle-Satisfying Third Order Discontinuous Galerkin Schemes for Fokker-Planck Equations

- Mathematics, Computer Science
- SIAM J. Sci. Comput.
- 2014

It is shown that a modified limiter can preserve the strict maximum principle for DG schemes solving Fokker--Planck equations, and a scaling limiter for the DG method with first order Euler forward time discretization is proposed to solve the one-dimensional Fokkersonian equations. Expand

Stability and Convergence Analysis of Fully Discrete Fourier Collocation Spectral Method for 3-D Viscous Burgers’ Equation

- Mathematics, Computer Science
- J. Sci. Comput.
- 2012

The main result in this work is that the pseudospectral method coupled with the carefully designed time-discretizations is stable provided only that the time-step and spatial grid-size are bounded by two constants over a finite time. Expand