The Experts below are selected from a list of 76005 Experts worldwide ranked by ideXlab platform
Jiang Yang - One of the best experts on this subject based on the ideXlab platform.
-
arbitrarily high order exponential cut off methods for preserving Maximum Principle of parabolic equations
SIAM Journal on Scientific Computing, 2020Co-Authors: Jiang Yang, Zhi ZhouAbstract:A new class of high-order Maximum Principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear Allen--Cahn equation. The proposed method co...
-
arbitrarily high order exponential cut off methods for preserving Maximum Principle of parabolic equations
arXiv: Numerical Analysis, 2020Co-Authors: Jiang Yang, Zhi ZhouAbstract:A new class of high-order Maximum Principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear Allen--Cahn equation. The proposed method consists of a $k$th-order multistep exponential integrator in time, and a lumped mass finite element method in space with piecewise $r$th-order polynomials and Gauss--Lobatto quadrature. At every time level, the extra values violating the Maximum Principle are eliminated at the finite element nodal points by a cut-off operation. The remaining values at the nodal points satisfy the Maximum Principle and are proved to be convergent with an error bound of $O(\tau^k+h^r)$. The accuracy can be made arbitrarily high-order by choosing large $k$ and $r$. Extensive numerical results are provided to illustrate the accuracy of the proposed method and the effectiveness in capturing the pattern of phase-field problems.
-
implicit explicit scheme for the allen cahn equation preserves the Maximum Principle
Journal of Computational Mathematics, 2016Co-Authors: Tao Tang, Jiang YangAbstract:It is known that the Allen-Chan equations satisfy the Maximum Principle. Is this true for numerical schemes? To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the Maximum Principle.
-
on the Maximum Principle preserving schemes for the generalized allen cahn equation
Communications in Mathematical Sciences, 2016Co-Authors: Jie Shen, Tao Tang, Jiang YangAbstract:This paper is concerned with the generalized Allen-Cahn equation with a nonlinear mobility that may be degenerate, which also includes an advection term appearing in many phase- field models for multi-component fluid flows. A class of Maximum Principle preserving schemes will be studied for the generalized Allen-Cahn equation, with either the commonly used polynomial free energy or the logarithmic free energy, and with a nonlinear degenerate mobility. For time discretization, the standard semi-implicit scheme as well as the stabilized semi-implicit scheme will be adopted, while for space discretization, the central finite difference is used for approximating the diffusion term and the upwind scheme is employed for the advection term. We establish the Maximum Principle for both semi-discrete (in time) and fully discretized schemes. We also provide an error estimate by using the established Maximum Principle which plays a key role in the analysis. Several numerical experiments are carried out to verify our theoretical results.
Xiangxiong Zhang - One of the best experts on this subject based on the ideXlab platform.
-
on Maximum Principle satisfying high order schemes for scalar conservation laws
Journal of Computational Physics, 2010Co-Authors: Xiangxiong ZhangAbstract:We construct uniformly high order accurate schemes satisfying a strict Maximum Principle for scalar conservation laws. A general framework (for arbitrary order of accuracy) is established to construct a limiter for finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or discontinuous Galerkin (DG) method with first order Euler forward time discretization solving one-dimensional scalar conservation laws. Strong stability preserving (SSP) high order time discretizations will keep the Maximum Principle. It is straightforward to extend the method to two and higher dimensions on rectangular meshes. We also show that the same limiter can preserve the Maximum Principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. Numerical tests for both the WENO finite volume scheme and the DG method are reported.
Zhonghua Qiao - One of the best experts on this subject based on the ideXlab platform.
-
Maximum Principle preserving exponential time differencing schemes for the nonlocal allen cahn equation
SIAM Journal on Numerical Analysis, 2019Co-Authors: Zhonghua QiaoAbstract:The nonlocal Allen--Cahn equation, a generalization of the classic Allen--Cahn equation by replacing the Laplacian with a parameterized nonlocal diffusion operator, satisfies the Maximum Principle ...
-
Maximum Principle preserving exponential time differencing schemes for the nonlocal allen cahn equation
arXiv: Numerical Analysis, 2019Co-Authors: Zhonghua QiaoAbstract:The nonlocal Allen-Cahn (NAC) equation is a generalization of the classic Allen-Cahn equation by replacing the Laplacian with a parameterized nonlocal diffusion operator, and satisfies the Maximum Principle as its local counterpart. In this paper, we develop and analyze first and second order exponential time differencing (ETD) schemes for solving the NAC equation, which unconditionally preserve the discrete Maximum Principle. The fully discrete numerical schemes are obtained by applying the stabilized ETD approximations for time integration with the quadrature-based finite difference discretization in space. We derive their respective optimal Maximum-norm error estimates and further show that the proposed schemes are asymptotically compatible, i.e., the approximate solutions always converge to the classic Allen-Cahn solution when the horizon, the spatial mesh size and the time step size go to zero. We also prove that the schemes are energy stable in the discrete sense. Various experiments are performed to verify these theoretical results and to investigate numerically the relation between the discontinuities and the nonlocal parameters.
Masahiro Yamamoto - One of the best experts on this subject based on the ideXlab platform.
-
on the Maximum Principle for the multi term fractional transport equation
arXiv: Analysis of PDEs, 2021Co-Authors: Yuri Luchko, Anna Suzuki, Masahiro YamamotoAbstract:In this paper, we prove a Maximum Principle for the general multi-term space-time-fractional transport equation and apply it for establishing uniqueness of solution to an initial-boundary-value problem for this equation. We also derive some comparison Principles for solutions to the initial-boundary-value problems with different problem data. Finally, we present a Maximum Principle for the Cauchy problem for a time-fractional transport equation on an unbounded domain.
-
on the Maximum Principle for a time fractional diffusion equation
arXiv: Analysis of PDEs, 2017Co-Authors: Yuri Luchko, Masahiro YamamotoAbstract:In this paper, we discuss the Maximum Principle for a time-fractional diffusion equation $$ \partial_t^\alpha u(x,t) = \sum_{i,j=1}^n \partial_i(a_{ij}(x)\partial_j u(x,t)) + c(x)u(x,t) + F(x,t),\ t>0,\ x \in \Omega \subset {\mathbb R}^n$$ with the Caputo time-derivative of the order $\alpha \in (0,1)$ in the case of the homogeneous Dirichlet boundary condition. Compared to the already published results, our findings have two important special features. First, we derive a Maximum Principle for a suitably defined weak solution in the fractional Sobolev spaces, not for the strong solution. Second, for the non-negative source functions $F = F(x,t)$ we prove the non-negativity of the weak solution to the problem under consideration without any restrictions on the sign of the coefficient $c=c(x)$ by the derivative of order zero in the spatial differential operator. Moreover, we prove the monotonicity of the solution with respect to the coefficient $c=c(x)$.
-
strong Maximum Principle for fractional diffusion equations and an application to an inverse source problem
arXiv: Analysis of PDEs, 2015Co-Authors: Yikan Liu, William Rundell, Masahiro YamamotoAbstract:The strong Maximum Principle is a remarkable characterization of parabolic equations, which is expected to be partly inherited by fractional diffusion equations. Based on the corresponding weak Maximum Principle, in this paper we establish a strong Maximum Principle for time-fractional diffusion equations with Caputo derivatives, which is slightly weaker than that for the parabolic case. As a direct application, we give a uniqueness result for a related inverse source problem on the determination of the temporal component of the inhomogeneous term.
Zhi Zhou - One of the best experts on this subject based on the ideXlab platform.
-
arbitrarily high order exponential cut off methods for preserving Maximum Principle of parabolic equations
SIAM Journal on Scientific Computing, 2020Co-Authors: Jiang Yang, Zhi ZhouAbstract:A new class of high-order Maximum Principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear Allen--Cahn equation. The proposed method co...
-
arbitrarily high order exponential cut off methods for preserving Maximum Principle of parabolic equations
arXiv: Numerical Analysis, 2020Co-Authors: Jiang Yang, Zhi ZhouAbstract:A new class of high-order Maximum Principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear Allen--Cahn equation. The proposed method consists of a $k$th-order multistep exponential integrator in time, and a lumped mass finite element method in space with piecewise $r$th-order polynomials and Gauss--Lobatto quadrature. At every time level, the extra values violating the Maximum Principle are eliminated at the finite element nodal points by a cut-off operation. The remaining values at the nodal points satisfy the Maximum Principle and are proved to be convergent with an error bound of $O(\tau^k+h^r)$. The accuracy can be made arbitrarily high-order by choosing large $k$ and $r$. Extensive numerical results are provided to illustrate the accuracy of the proposed method and the effectiveness in capturing the pattern of phase-field problems.
