In this paper, we consider a numerical method for the multi-term Caputo-Fabrizio time-fractional diffusion equations (with orders $\alpha_i\in(0,1)$, $i=1,2,\cdots,n$). The proposed method employs a fast finite difference scheme to approximate multi-term fractional derivatives in time, requiring only $O(1)$ storage and $O(N_T)$ computational complexity, where $N_T$ denotes the total number of time steps. Then we use a Legendre spectral collocation method for spatial discretization. The stability and convergence of the scheme have been thoroughly discussed and rigorously established. We demonstrate that the proposed scheme is unconditionally stable and convergent with an order of $O(\left(\Delta t\right)^{2}+N^{-m})$, where $\Delta t$, $N$, and $m$ represent the timestep size, polynomial degree, and regularity in the spatial variable of the exact solution, respectively. Numerical results are presented to validate the theoretical predictions.
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泛函
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It is disproved the Tokareva's conjecture that any balanced boolean function of appropriate degree is a derivative of some bent function. This result is based on new upper bounds for the numbers of bent and plateaued functions.
The semi-empirical nature of best-estimate models closing the balance equations of thermal-hydraulic (TH) system codes is well-known as a significant source of uncertainty for accuracy of output predictions. This uncertainty, called model uncertainty, is usually represented by multiplicative (log-)Gaussian variables whose estimation requires solving an inverse problem based on a set of adequately chosen real experiments. One method from the TH field, called CIRCE, addresses it. We present in the paper a generalization of this method to several groups of experiments each having their own properties, including different ranges for input conditions and different geometries. An individual (log-)Gaussian distribution is therefore estimated for each group in order to investigate whether the model uncertainty is homogeneous between the groups, or should depend on the group. To this end, a multi-group CIRCE is proposed where a variance parameter is estimated for each group jointly to a mean parameter common to all the groups to preserve the uniqueness of the best-estimate model. The ECME algorithm for Maximum Likelihood Estimation is adapted to the latter context, then applied to relevant demonstration cases. Finally, it is tested on a practical case to assess the uncertainty of critical mass flow assuming two groups due to the difference of geometry between the experimental setups.
For a singular integral equation on an interval of the real line, we study the behavior of the error of a delta-delta discretization. We show that the convergence is non-uniform, between order $O(h^{2})$ in the interior of the interval and a boundary layer where the consistency error does not tend to zero.
In this paper we establish limit theorems for power variations of stochastic processes controlled by fractional Brownian motions with Hurst parameter $H\leq 1/2$. We show that the power variations of such processes can be decomposed into the mix of several weighted random sums plus some remainder terms, and the convergences of power variations are dominated by different combinations of those weighted sums depending on whether $H<1/4$, $H=1/4$, or $H>1/4$. We show that when $H\geq 1/4$ the centered power variation converges stably at the rate $n^{-1/2}$, and when $H<1/4$ it converges in probability at the rate $n^{-2H}$. We determine the limit of the mixed weighted sum based on a rough path approach developed in \cite{LT20}.
In 1-equation URANS models of turbulence the eddy viscosity is given by $\nu_{T}=0.55l(x,t)\sqrt{k(x,t)}$ . The length scale $l$ must be pre-specified and $k(x,t)$ is determined by solving a nonlinear partial differential equation. We show that in interesting cases the spacial mean of $k(x,t)$ satisfies a simple ordinary differential equation. Using its solution in $\nu_{T}$ results in a 1/2-equation model. This model has attractive analytic properties. Further, in comparative tests in 2d and 3d the velocity statistics produced by the 1/2-equation model are comparable to those of the full 1-equation model.
The nonlocality of the fractional operator causes numerical difficulties for long time computation of the time-fractional evolution equations. This paper develops a high-order fast time-stepping discontinuous Galerkin finite element method for the time-fractional diffusion equations, which saves storage and computational time. The optimal error estimate $O(N^{-p-1} + h^{m+1} + \varepsilon N^{r\alpha})$ of the current time-stepping discontinuous Galerkin method is rigorous proved, where $N$ denotes the number of time intervals, $p$ is the degree of polynomial approximation on each time subinterval, $h$ is the maximum space step, $r\ge1$, $m$ is the order of finite element space, and $\varepsilon>0$ can be arbitrarily small. Numerical simulations verify the theoretical analysis.
This paper introduces a formulation of the variable density incompressible Navier-Stokes equations by modifying the nonlinear terms in a consistent way. For Galerkin discretizations, the formulation leads to full discrete conservation of mass, squared density, momentum, angular momentum and kinetic energy without the divergence-free constraint being strongly enforced. In addition to favorable conservation properties, the formulation is shown to make the density field invariant to global shifts. The effect of viscous regularizations on conservation properties is also investigated. Numerical tests validate the theory developed in this work. The new formulation shows superior performance compared to other formulations from the literature, both in terms of accuracy for smooth problems and in terms of robustness.
In this paper, we study functional approximations where we choose the so-called radial basis function method and more specifically, quasi-interpolation. From the various available approaches to the latter, we form new quasi-Lagrange functions when the orders of the singularities of the radial function's Fourier transforms at zero do not match the parity of the dimension of the space, and therefore new expansions and coefficients are needed to overcome this problem. We develop explicit constructions of infinite Fourier expansions that provide these coefficients and make an extensive comparison of the approximation qualities and - with a particular focus - polynomial precision and uniform approximation order of the various formulae. One of the interesting observations concerns the link between algebraic conditions of expansion coefficients and analytic properties of localness and convergence.
In this work, a Cole-Hopf transformation based fourth-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for d-dimensional coupled Burgers' equations is developed. We first adopt the Cole-Hopf transformation where an intermediate variable \theta is introduced to eliminate the nonlinear convection terms in the Burgers' equations on the velocity u=(u_1,u_2,...,u_d). In this case, a diffusion equation on the variable \theta can be obtained, and particularly, the velocity u in the coupled Burgers' equations is determined by the variable \theta and its gradient term \nabla\theta. Then we develop a general MRT-LB model with the natural moments for the d-dimensional transformed diffusion equation and present the corresponding macroscopic finite-difference scheme. At the diffusive scaling, the fourth-order modified equation of the developed MRT-LB model is derived through the Maxwell iteration method. With the aid of the free parameters in the MRT-LB model, we find that not only the consistent fourth-order modified equation can be obtained, but also the gradient term $\nabla\theta$ can be calculated locally by the non-equilibrium distribution function with a fourth-order accuracy, this indicates that theoretically, the MRT-LB model for $d$-dimensional coupled Burgers' equations can achieve a fourth-order accuracy in space. Finally, some simulations are conducted to test the MRT-LB model, and the numerical results show that the proposed MRT-LB model has a fourth-order convergence rate, which is consistent with our theoretical analysis.
In this paper, we are concerned with symmetric integrators for the nonlinear relativistic Klein--Gordon (NRKG) equation with a dimensionless parameter $0<\varepsilon\ll 1$, which is inversely proportional to the speed of light. The highly oscillatory property in time of this model corresponds to the parameter $\varepsilon$ and the equation has strong nonlinearity when $\eps$ is small. There two aspects bring significantly numerical burdens in designing numerical methods. We propose and analyze a novel class of symmetric integrators which is based on some formulation approaches to the problem, Fourier pseudo-spectral method and exponential integrators. Two practical integrators up to order four are constructed by using the proposed symmetric property and stiff order conditions of implicit exponential integrators. The convergence of the obtained integrators is rigorously studied, and it is shown that the accuracy in time is improved to be $\mathcal{O}(\varepsilon^{3} \hh^2)$ and $\mathcal{O}(\varepsilon^{4} \hh^4)$ for the time stepsize $\hh$. The near energy conservation over long times is established for the multi-stage integrators by using modulated Fourier expansions. These theoretical results are achievable even if large stepsizes are utilized in the schemes. Numerical results on a NRKG equation show that the proposed integrators have improved uniform error bounds, excellent long time energy conservation and competitive efficiency.
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