In this paper, we design and analyze a space-time spectral method for the subdiffusion equation.Here, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the init...In this paper, we design and analyze a space-time spectral method for the subdiffusion equation.Here, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the initial time. Consequently, traditional high-order numerical methods in time are inefficient. The second obstacle is that the resulting system of the space-time spectral approach is usually large and time-consuming to solve. We aim at overcoming the first difficulty by proposing a novel approach in time, which is based on variable transformation techniques. Suitable ψ-fractional Sobolev spaces and a new variational framework are introduced to establish the well-posedness of the associated variational problem. This allows us to construct our space-time spectral method using a combination of temporal generalized Jacobi polynomials(GJPs) and spatial Legendre polynomials. For the second difficulty, we propose a fast algorithm to effectively solve the resulting linear system. The fast algorithm makes use of a matrix diagonalization in space and QZ decomposition in time. Our analysis and numerical experiments show that the proposed method is exponentially convergent with respect to the polynomial degrees in both space and time directions, even though the exact solution has very limited regularity.展开更多
An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial time.The weak regularity of the solution is usually characterized by a regularity p...An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial time.The weak regularity of the solution is usually characterized by a regularity parameterσ∈(0,1)∪(1,2).Under this general regularity assumption,we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results,i.e.,a refined discrete fractional-type Grönwall inequality(DFGI).After that,we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations.The present results fill the gap on some interesting convergence results of L1 scheme onσ∈(0,α)∪(α,1)∪(1,2].Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.展开更多
In this paper,we study a high-order compact difference scheme for the fourth-order fractional subdiffusion system.We consider the situation in which the unknown function and its first-order derivative are given at the...In this paper,we study a high-order compact difference scheme for the fourth-order fractional subdiffusion system.We consider the situation in which the unknown function and its first-order derivative are given at the boundary.The scheme is shown to have high order convergence.Numerical examples are given to verify the theoretical results.展开更多
We present the finite difference/element method for a two-dimensional modified fractional diffusion equation.The analysis is carried out first for the time semi-discrete scheme,and then for the full discrete scheme.Th...We present the finite difference/element method for a two-dimensional modified fractional diffusion equation.The analysis is carried out first for the time semi-discrete scheme,and then for the full discrete scheme.The time discretization is based on the L1-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term.We use finite element method for the spatial approximation in full discrete scheme.We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent.Moreover,the optimal convergence rate is obtained.Finally,some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 11971408)。
文摘In this paper, we design and analyze a space-time spectral method for the subdiffusion equation.Here, we are facing two difficulties. The first is that the solutions of this equation are usually singular near the initial time. Consequently, traditional high-order numerical methods in time are inefficient. The second obstacle is that the resulting system of the space-time spectral approach is usually large and time-consuming to solve. We aim at overcoming the first difficulty by proposing a novel approach in time, which is based on variable transformation techniques. Suitable ψ-fractional Sobolev spaces and a new variational framework are introduced to establish the well-posedness of the associated variational problem. This allows us to construct our space-time spectral method using a combination of temporal generalized Jacobi polynomials(GJPs) and spatial Legendre polynomials. For the second difficulty, we propose a fast algorithm to effectively solve the resulting linear system. The fast algorithm makes use of a matrix diagonalization in space and QZ decomposition in time. Our analysis and numerical experiments show that the proposed method is exponentially convergent with respect to the polynomial degrees in both space and time directions, even though the exact solution has very limited regularity.
基金supported by the National Natural Science Foundation of China under grants 11771162,11771035,12171376 and 2020-JCJQ-ZD-029.
文摘An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial time.The weak regularity of the solution is usually characterized by a regularity parameterσ∈(0,1)∪(1,2).Under this general regularity assumption,we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results,i.e.,a refined discrete fractional-type Grönwall inequality(DFGI).After that,we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations.The present results fill the gap on some interesting convergence results of L1 scheme onσ∈(0,α)∪(α,1)∪(1,2].Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.
基金the Macao Science and Technology Development Fund FDCT/001/2013/A and the grant MYRG086(Y2-L2)-FST12-VSW from the University of Macao.
文摘In this paper,we study a high-order compact difference scheme for the fourth-order fractional subdiffusion system.We consider the situation in which the unknown function and its first-order derivative are given at the boundary.The scheme is shown to have high order convergence.Numerical examples are given to verify the theoretical results.
基金This research was partly supported by the National Basic Research Program of China973 Program under Grant No.2011CB706903+3 种基金the Program for New Century Excellent Talents in University under Grant No.NCET-09-0438the National Natural Science Foundation of China under Grant No.10801067the Fundamental Research Funds for the Central Universities under Grant No.lzujbky-2010-63No.lzujbky-2012-k26。
文摘We present the finite difference/element method for a two-dimensional modified fractional diffusion equation.The analysis is carried out first for the time semi-discrete scheme,and then for the full discrete scheme.The time discretization is based on the L1-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term.We use finite element method for the spatial approximation in full discrete scheme.We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent.Moreover,the optimal convergence rate is obtained.Finally,some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.