A Space–Time Spectral Method for Nonlinear Fractional Convection–Diffusion Equations with Viscosity Terms

A semi-analytical Fourier spectral method for the Allen–Cahn equation

Generalized polynomial chaos for the convection diffusion equation with uncertainty

In this paper we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations(MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation(NACDE) and Navier-Stokes equations(NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme. In particular, for the one-dimensional convection-diffusion equation(CDE) with the constant velocity and diffusion coefficient, we can develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, three benchmark problems, the Gauss hill problem, the CDE with nonlinear convection and diffusion terms, and the Taylor-Green vortex flow in two-dimensional space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found that the numerical results not only are in good agreement with corresponding analytical solutions, but also have a second-order convergence rate in space. Additionally, a numerical study on one-dimensional CDE also demonstrates that the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in space, which is consistent with our theoretical analysis.

Kink solutions of the generalized, super-diffusive Burgers equation with memory

An efficient numerical method for the modified regularized long wave equation using Fourier spectral method

Spectral method for solving the time fractional Boussinesq equation

Standard Fourier spectral method can be used to solve a lot of problems with periodic boundary conditions. However, for non-periodic boundary condition problems, standard Fourier spectral method is not efficient or even useless. This work has used the modified Fourier spectral method for non-periodic boundary condition problems. The modified spectral method has two parts, first, the original function is normalized and then a smooth buffer polynomial is developed to extend the normalized function domain. The new function will be smooth and periodic with both function values and derivatives, which is easy to be treated by standard FFT for high resolution. This method has obtained high order accuracy and high resolution with a penalty of 25% over standard Fourier spectral method, as shown by our examples. The scheme demonstrates to be robust. The current examples include a 2D Poisson solver and solution for 2-D incompressible driven cavity flow. The method will be further used for simulation of transitional and turbulent flow.

IIM-based ADI finite difference scheme for nonlinear convection–diffusion equations with interfaces

We deal with the numerical solution of a scalar nonstationary nonlinear convection–diffusion equation. We employ a combination of the discontinuous Galerkin finite element method for the space semi-discretization and the k-step backward difference formula for the time discretization. The diffusive and stabilization terms are treated implicitly whereas the nonlinear convective term is treated by a higher order explicit extrapolation method, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the discrete L ∞(L 2)-norm and the L 2(H 1)-seminorm with respect to the mesh size h and time step τ for k = 2,3. Numerical examples verifying the theoretical results are presented.

Analysis of semi-implicit DGFEM for nonlinear convection–diffusion problems on nonconforming meshes

A finite difference method (FDM) is easier and cheaper to use than a spectral method (SM) in flows with complicated geometry, but is inferior to a SM in both accuracy and resolution. In this paper, effects of both accuracy and resolution in a FDM on the quality of incompressible turbulent flow description were studied by comparing a SM with a FDM in direct numerical simulations (DNS) of the three-dimensional freely decaying homogeneous turbulence with 1283 and 2563 grids. We adopted the higher-order method of lines and Fourier spectral method as a FDM and a SM, respectively. The value of the kinematic viscosity was determined so that the minimum value of Kmaxη in time was about 2 in the SM with 2563 grids. Here, Kmax and η are the maximum resolved wavenumber and the Kolmogorov length scale, respectively. The statistical quantities, such as the total energy, the energy dissipation ratio, obtained by the higher-order method of lines were in good agreement with those by the SM. Furthermore, comparing their DNS fields, we found that the performance of the 4th-order method of lines with 2563 grids was comparable to that of the SM with 1283 grids in which the minimum value of Kmaxη in time was about 1.

A generalized-Jacobi-function spectral method for space-time fractional reaction-diffusion equations with viscosity terms

In this article, we are devoted to studying the following nonlocal Navier–Stokes equations involving the time–space fractional operators { ∂ t β u + ( − Δ ) α u + u ⋅ ∇ u = ∇ p + λ | u | q − 2 u + g ( x , t ) in Ω × R + , div u = 0 in Ω × R + , u ( x , t ) = 0 in ( R N ∖ Ω ) × R + , u ( x , 0 ) = u 0 ( x ) in Ω , where Ω ⊂ R N is a bounded domain with Lipschitz boundary, ( − Δ ) α is the fractional Laplacian with 0 < α < 1 , ∂ t β is the Riemann–Liouville time fractional derivative with 0 < β < 1 , λ is a parameter, and g ∈ ( L ∞ ( 0 , ∞ ; L 2 ( Ω ) ) ) N . Due to the nonlocal nature of the fractional Laplacian, the classical weak solution frameworks proposed by J. L. Lions and E. Hopf are inapplicable to this problem. To address this challenge, we first construct a regularized problem and establish the existence of its solutions via the Galerkin method combined with fractional calculus techniques. Subsequently, we prove the uniqueness of the global weak solution to the regularized problem. Furthermore, under appropriate assumptions, a decay estimate characterizing the long-time behavior of solutions is derived. Finally, the existence of solutions to the original problem is obtained by analyzing the limiting behavior of the regularized solutions as the regularization parameter vanishes. The main contributions of this work are twofold: (1) Our problem simultaneously incorporates the Riemann–Liouville time fractional derivative and the fractional Laplacian, extending classical Navier–Stokes theory to a nonlocal time–space framework. (2) We develop a new analytical approach to estimate the nonlinear convection term u ⋅ ∇ u in the presence of fractional operators, overcoming challenges posed by the loss of localization.

A Level-Set Method for Computing Solutions to Viscoelastic Two-Phase Flow

In this paper, we study thetime-asymptotic behavior of the solution for the Cauchy problem ofthe damped wave equation with a nonlinear convection term in themulti-dimensional space. When the initial data is a small perturbation around aconstant state $u^*$, we obtain the point-wise decay estimates of thesolution under theso-called dissipative condition $|b| < 1$, where $b$ depends on $u^*$ and the nonlinear term.