The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles. The related governing equations are a series of partial differential equations with tempered fractional derivatives. Using the polynomial interpolation technique, in this paper, we present three efficient numerical formulas, namely the tempered L1 formula, the tempered L1-2 formula, and the tempered L2-$1_{\sigma }$ formula, to approximate the Caputo-tempered fractional derivative of order $\alpha \in (0,1)$. The truncation error of the tempered L1 formula is of order 2−$\alpha$, and the tempered L1-2 formula and L2-$1_{\sigma }$ formula are of order 3−$\alpha$. As an application, we construct implicit schemes and implicit ADI schemes for one-dimensional and two-dimensional time-tempered fractional diffusion equations, respectively. Furthermore, the unconditional stability and convergence of two developed difference schemes with tempered L1 and L2-$1_{\sigma }$ formulas are proved by the Fourier analysis method. Finally, we provide several numerical examples to demonstrate the correctness and effectiveness of the theoretical analysis.
By transforming the Caputo tempered fractional advection-diffusion equation into the Riemann–Liouville tempered fractional advection-diffusion equation, and then using the fractional-compact Grünwald–Letnikov tempered difference operator to approximate the Riemann–Liouville tempered fractional partial derivative, the fractional central difference operator to discritize the space Riesz fractional partial derivative, and the classical central difference formula to discretize the advection term, a numerical algorithm is constructed for solving the Caputo tempered fractional advection-diffusion equation. The stability and the convergence analysis of the numerical method are given. Numerical experiments show that the numerical method is effective.
Fractional calculus and fractional-order modeling provide effective tools for modeling and simulation of anomalous diffusion with power-law scalings. In complex multi-fractal anomalous transport phenomena, distributed-order partial differential equations appear as tractable mathematical models, where the underlying derivative orders are distributed over a range of values, hence taking into account a wide range of multi-physics from ultraslow-to-standard-to-superdiffusion/wave dynamics. We develop a unified, fast, and stable Petrov–Galerkin spectral method for such models by employing Jacobi poly-fractonomials and Legendre polynomials as temporal and spatial basis/test functions, respectively. By defining the proper underlying distributed Sobolev spaces and their equivalent norms, we rigorously prove the well-posedness of the weak formulation, and thereby, we carry out the corresponding stability and error analysis. We finally provide several numerical simulations to study the performance and convergence of proposed scheme.
A modified weak Galerkin (MWG) finite element method is developed for solving the biharmonic equation. This method uses the same finite element space as that of the discontinuous Galerkin method, the space of discontinuous polynomials on polytopal meshes. But its formulation is simple, symmetric, positive definite, and parameter independent, without any of six inter-element face-integral terms in the formulation of the discontinuous Galerkin method. Optimal order error estimates in a discrete $H^2$ norm are established for the corresponding finite element solutions. Error estimates in the $L^2$ norm are also derived with a sub-optimal order of convergence for the lowest-order element and an optimal order of convergence for all high-order of elements. The numerical results are presented to confirm the theory of convergence.
In this paper, we study the nonlinear matrix equation $X-{A^{\rm{H}}}{\overline{X}}^{-1}A=Q$, where $A,Q \in {{\mathbb {C}}}^{n\times n}$, Q is a Hermitian positive definite matrix and $X \in {{\mathbb {C}}}^{n\times n}$ is an unknown matrix. We prove that the equation always has a unique Hermitian positive definite solution. We present two structure-preserving-doubling like algorithms to find the Hermitian positive definite solution of the equation, and the convergence theories are established. Finally, we show the effectiveness of the algorithms by numerical experiments.
The generalized singular value decomposition (GSVD) of two matrices with the same number of columns is a very useful tool in many practical applications. However, the GSVD may suffer from heavy computational time and memory requirement when the scale of the matrices is quite large. In this paper, we use random projections to capture the most of the action of the matrices and propose randomized algorithms for computing a low-rank approximation of the GSVD. Serval error bounds of the approximation are also presented for the proposed randomized algorithms. Finally, some experimental results show that the proposed randomized algorithms can achieve a good accuracy with less computational cost and storage requirement.
After discretization by the finite volume method, the numerical solution of fractional diffusion equations leads to a linear system with the Toeplitz-like structure. The theoretical analysis gives sufficient conditions to guarantee the positive-definite property of the discretized matrix. Moreover, we develop a class of positive-definite operator splitting iteration methods for the numerical solution of fractional diffusion equations, which is unconditionally convergent for any positive constant. Meanwhile, the iteration methods introduce a new preconditioner for Krylov subspace methods. Numerical experiments verify the convergence of the positive-definite operator splitting iteration methods and show the efficiency of the proposed preconditioner, compared with the existing approaches.
In this paper, for the regularized Hermitian and skew-Hermitian splitting (RHSS) preconditioner introduced by Bai and Benzi (BIT Numer Math 57: 287–311, 2017) for the solution of saddle-point linear systems, we analyze the spectral properties of the preconditioned matrix when the regularization matrix is a special Hermitian positive semidefinite matrix which depends on certain parameters. We accurately describe the numbers of eigenvalues clustered at (0, 0) and (2, 0), if the iteration parameter is close to 0. An estimate about the condition number of the corresponding eigenvector matrix, which partly determines the convergence rate of the RHSS-preconditioned Krylov subspace method, is also studied in this work.
Bai et al. proposed the multistep Rayleigh quotient iteration (MRQI) as well as its inexact variant (IMRQI) in a recent work (Comput. Math. Appl. 77: 2396–2406, 2019). These methods can be used to effectively compute an eigenpair of a Hermitian matrix. The convergence theorems of these methods were established under two conditions imposed on the initial guesses for the target eigenvalue and eigenvector. In this paper, we show that these two conditions can be merged into a relaxed one, so the convergence conditions in these theorems can be weakened, and the resulting convergence theorems are applicable to a broad class of matrices. In addition, we give detailed discussions about the new convergence condition and the corresponding estimates of the convergence errors, leading to rigorous convergence theories for both the MRQI and the IMRQI.