This paper aims to numerically study the generalized time-fractional Burgers equation in two spatial dimensions using the L1/LDG method. Here the L1 scheme is used to approximate the time-fractional derivative, i.e., Caputo derivative, while the local discontinuous Galerkin (LDG) method is used to discretize the spatial derivative. If the solution has strong temporal regularity, i.e., its second derivative with respect to time being right continuous, then the L1 scheme on uniform meshes (uniform L1 scheme) is utilized. If the solution has weak temporal regularity, i.e., its first and/or second derivatives with respect to time blowing up at the starting time albeit the function itself being right continuous at the beginning time, then the L1 scheme on non-uniform meshes (non-uniform L1 scheme) is applied. Then both uniform L1/LDG and non-uniform L1/LDG schemes are constructed. They are both numerically stable and the $L^2$ optimal error estimate for the velocity is obtained. Numerical examples support the theoretical analysis.
A stabilizer-free weak Galerkin (SFWG) finite element method was introduced and analyzed in Ye and Zhang (SIAM J. Numer. Anal. 58: 2572–2588, 2020) for the biharmonic equation, which has an ultra simple finite element formulation. This work is a continuation of our investigation of the SFWG method for the biharmonic equation. The new SFWG method is highly accurate with a convergence rate of four orders higher than the optimal order of convergence in both the energy norm and the $L^2$ norm on triangular grids. This new method also keeps the formulation that is symmetric, positive definite, and stabilizer-free. Four-order superconvergence error estimates are proved for the corresponding SFWG finite element solutions in a discrete $H^2$ norm. Superconvergence of four orders in the $L^2$ norm is also derived for $k\geqslant 3$, where k is the degree of the approximation polynomial. The postprocessing is proved to lift a $P_k$ SFWG solution to a $P_{k+4}$ solution elementwise which converges at the optimal order. Numerical examples are tested to verify the theories.
High-order accurate weighted essentially non-oscillatory (WENO) schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations (PDEs). Due to highly nonlinear property of the WENO algorithm, large amount of computational costs are required for solving multidimensional problems. In our previous work (Lu et al. in Pure Appl Math Q 14: 57–86, 2018; Zhu and Zhang in J Sci Comput 87: 44, 2021), sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations, and it was shown that significant CPU times were saved, while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids. In this technical note, we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme, which has very interesting properties such as its simplicity in linear weights’ construction over a classical WENO scheme. Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times, and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.
The work proposes a model of biological fluid flow in a steady mode through a cylindrical layer taking into account convection and diffusion. The model considers finite compressibility and concentration expansion connected with both barodiffusion and additional mechanism of pressure change in the pore volume due to the concentration gradient. Thus, the model is entirely coupled. The paper highlights the complexes composed of scales of physical quantities of different natures. The iteration algorithm for the numerical solution of the problem was developed for the coupled problem. The work involves numerical studies of the considered effects on the characteristics of the flow that can be convective or diffusive, depending on the relation between the dimensionless complexes. It is demonstrated that the distribution of velocity and concentration for different cylinder wall thicknesses is different. It is established that the barodiffusion has a considerable impact on the process in the convective mode or in the case of reduced cylinder wall thickness.
Fourier continuation (FC) is an approach used to create periodic extensions of non-periodic functions to obtain highly-accurate Fourier expansions. These methods have been used in partial differential equation (PDE)-solvers and have demonstrated high-order convergence and spectrally accurate dispersion relations in numerical experiments. Discontinuous Galerkin (DG) methods are increasingly used for solving PDEs and, as all Galerkin formulations, come with a strong framework for proving the stability and the convergence. Here we propose the use of FC in forming a new basis for the DG framework.
In this paper, the weak pre-orthogonal adaptive Fourier decomposition (W-POAFD) method is applied to solve fractional boundary value problems (FBVPs) in the reproducing kernel Hilbert spaces (RKHSs) $W^{4}_0[0,1]$ and $W^1[0,1]$. The process of the W-POAFD is as follows: (i) choose a dictionary and implement the pre-orthogonalization to all the dictionary elements; (ii) select points in [0, 1] by the weak maximal selection principle to determine the corresponding orthonormalized dictionary elements iteratively; (iii) express the analytical solution as a linear combination of these determined dictionary elements. Convergence properties of numerical solutions are also discussed. The numerical experiments are carried out to illustrate the accuracy and efficiency of W-POAFD for solving FBVPs.
In this paper, a two-step semi-regularized Hermitian and skew-Hermitian splitting (SHSS) iteration method is constructed by introducing a regularization matrix in the (1,1)-block of the first iteration step, to solve the saddle-point linear system. By carefully selecting two different regularization matrices, two kinds of SHSS preconditioners are proposed to accelerate the convergence rates of the Krylov subspace iteration methods. Theoretical analysis about the eigenvalue distribution demonstrates that the proposed SHSS preconditioners can make the eigenvalues of the corresponding preconditioned matrices be clustered around 1 and uniformly bounded away from 0. The eigenvector distribution and the upper bound on the degree of the minimal polynomial of the SHSS-preconditioned matrices indicate that the SHSS-preconditioned Krylov subspace iterative methods can converge to the true solution within finite steps in exact arithmetic. In addition, the numerical example derived from the optimal control problem shows that the SHSS preconditioners can significantly improve the convergence speeds of the Krylov subspace iteration methods, and their convergence rates are independent of the discrete mesh size.
In this paper, three kinds of discrete formulae for the Caputo fractional derivative are studied, including the modified L1 discretisation for $\alpha \in (0,1)$, and L2 discretisation and L2C discretisation for $\alpha \in (1,2)$. The truncation error estimates and the properties of the coefficients of all these discretisations are analysed in more detail. Finally, the theoretical analyses are verified by the numerical examples.
Several common dual quaternion functions, such as the power function, the magnitude function, the 2-norm function, and the kth largest eigenvalue of a dual quaternion Hermitian matrix, are standard dual quaternion functions, i.e., the standard parts of their function values depend upon only the standard parts of their dual quaternion variables. Furthermore, the sum, product, minimum, maximum, and composite functions of two standard dual functions, the logarithm and the exponential of standard unit dual quaternion functions, are still standard dual quaternion functions. On the other hand, the dual quaternion optimization problem, where objective and constraint function values are dual numbers but variables are dual quaternions, naturally arises from applications. We show that to solve an equality constrained dual quaternion optimization (EQDQO) problem, we only need to solve two quaternion optimization problems. If the involved dual quaternion functions are all standard, the optimization problem is called a standard dual quaternion optimization problem, and some better results hold. Then, we show that the dual quaternion optimization problems arising from the hand-eye calibration problem and the simultaneous localization and mapping (SLAM) problem are equality constrained standard dual quaternion optimization problems.
In this paper, we investigate some properties of dual complex numbers, dual complex vectors, and dual complex matrices. First, based on the magnitude of the dual complex number, we study the Young inequality, the Hölder inequality, and the Minkowski inequality in the setting of dual complex numbers. Second, we define the p-norm of a dual complex vector, which is a nonnegative dual number, and show some related properties. Third, we study the properties of eigenvalues of unitary matrices and unitary triangulation of arbitrary dual complex matrices. In particular, we introduce the operator norm of dual complex matrices induced by the p-norm of dual complex vectors, and give expressions of three important operator norms of dual complex matrices.
In this paper, we first initialize the S-product of tensors to unify the outer product, contractive product, and the inner product of tensors. Then, we introduce the separable symmetry tensors and separable anti-symmetry tensors, which are defined, respectively, as the sum and the algebraic sum of rank-one tensors generated by the tensor product of some vectors. We offer a class of tensors to achieve the upper bound for $\texttt {rank}({\mathcal {A}}) \leqslant 6$ for all tensors of size $3\times 3\times 3$. We also show that each $3\times 3\times 3$ anti-symmetric tensor is separable.
The paper is devoted to non-homogeneous second-order differential equations with polynomial right parts and polynomial coefficients. We derive estimates for the partial sums and products of the zeros of solutions to the considered equations. These estimates give us bounds for the function counting the zeros of solutions and information about the zero-free domains.
The existing randomized algorithms need an initial estimation of the tubal rank to compute an approximate tensor singular value decomposition. This paper proposes a new randomized fixed-precision algorithm which for a given third-order tensor and a prescribed approximation error bound, automatically finds the optimal tubal rank and corresponding low tubal rank approximation. The algorithm is based on the random projection technique and equipped with the power iteration method for achieving better accuracy. We conduct simulations on synthetic and real-world datasets to show the efficiency and performance of the proposed algorithm.
Gyllenberg and Yan (Discrete Contin Dyn Syst Ser B 11(2): 347–352, 2009) presented a system in Zeeman’s class 30 of 3-dimensional Lotka-Volterra (3D LV) competitive systems to admit at least two limit cycles, one of which is generated by the Hopf bifurcation and the other is obtained by the Poincaré-Bendixson theorem. Yu et al. (J Math Anal Appl 436: 521–555, 2016, Sect. 3.4) recalculated the first Liapunov coefficient of Gyllenberg and Yan’s system to be positive, rather than negative as in Gyllenberg and Yan (2009), and pointed out that the Poincaré-Bendixson theorem is not applicable for that system. Jiang et al. (J Differ Equ 284: 183–218, 2021, p. 213) proposed an open question: “whether Zeeman’s class 30 can be rigorously proved to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem?” This paper provides four systems in Zeeman’s class 30 to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem and gives an answer to the above question.
A finite difference/spectral scheme is proposed for the time fractional Ito equation. The mass conservation and stability of the numerical solution are deduced by the energy method in the $L^2$ norm form. To reduce the computation costs, the fast Fourier transform technic is applied to a pair of equivalent coupled differential equations. The effectiveness of the proposed algorithm is verified by the first numerical example. The mass conservation property and stability statement are confirmed by two other numerical examples.
This paper proposes a two-parameter block triangular splitting (TPTS) preconditioner for the general block two-by-two linear systems. The eigenvalues of the corresponding preconditioned matrix are proved to cluster around 0 or 1 under mild conditions. The limited numerical results show that the TPTS preconditioner is more efficient than the classic block-diagonal and block-triangular preconditioners when applied to the flexible generalized minimal residual (FGMRES) method.