This paper deals with a two-level factored Crank-Nicolson method in an approximate solution of two-dimensional evolutionary advection-diffusion equation with time dependent dispersion coefficients and sink/source terms subjects to appropriate initial and boundary conditions. The procedure consists of reducing problems in many space variables into a sequence of one-dimensional subproblems and then find the solution of tridiagonal linear systems of equations. This considerably reduces the computational cost of the algorithm. Furthermore, the proposed approach is fast and efficient: unconditionally stable, temporal second order accurate and fourth order convergent in space and it improves a large class of numerical schemes widely studied in the literature for the considered problem. The stability of the new method is deeply analyzed using the $L^{\infty}(t_{0},T_{f};L^{2})$-norm whereas the convergence rate of the scheme is numerically obtained in the $L^{2}$-norm. A broad range of numerical experiments are presented and critically discussed.

In this paper, we develop and analyze a discontinuous Galerkin (DG) method with minimal dissipation for the static bending problem of a finite-strain plate equation. The equations are deduced from a three-dimensional field equation. So the coupling of the equations and the mixed derivative terms are the barriers during developing discretization schemes. The error estimates of the scheme are proved in detail. Numerical experiments in different circumstances are presented to demonstrate the capabilities of the method.

In this paper, an efficient class of finite difference nested multi-resolution weighted essentially non-oscillatory (WENO) schemes with increasingly higher order of accuracy is presented for solving hyperbolic conservation laws on structured meshes. The crucial idea is originated from [44,46]. We only use the information defined on a series of nested unequal-sized central spatial stencils in high-order spatial approximations. These new nested multi-resolution WENO schemes use the same large central stencil as the same order classical WENO schemes [3,21] without increasing the number of the total spatial stencils, could obtain the optimal order of accuracy in smooth regions, and could control spurious oscillations nearby strong shocks or contact discontinuities by gradually degrading from ninth-order to seventh-order, fifth-order, third-order or ultimately to the first-order accuracy. Associated linear weights can be set as any positive numbers on condition that their summation is one. Therefore, these new WENO schemes are simple to construct and can be easily implemented to arbitrarily high-order accuracy in multi-dimensions. Some benchmark examples are illustrated to show the good performance of these new nested multi-resolution WENO schemes, especially in terms of robustness and low numerical dissipation.

We consider Anderson acceleration (AA) applied to two nonlinear solvers for the stationary Gross-Pitaevskii equation: a Picard type nonlinear iterative solver and a normalized gradient flow method. We formulate the solvers as fixed point problems and show that they both fit into the recently developed AA analysis framework. This allows us to prove that both methods' linear convergence rates are improved by a factor (less than one) from the gain of the AA optimization problem at each step. Numerical tests for finding ground state solutions in 1D and 2D show that AA significantly improves convergence behavior in both solvers, and additionally some comparisons between the solvers are drawn. A local convergence analysis for both methods are also provided.

Approximation of interacting kernels by sum of Gaussians (SOG) is frequently required in many applications of scientific and engineering computing in order to construct efficient algorithms for kernel summation or convolution problems. In this paper, we propose a kernel-independent SOG method by introducing the de la Vallée-Poussin sum and Chebyshev polynomials. The SOG works for general interacting kernels and the lower bound of Gaussian bandwidths is tunable and thus the Gaussians can be easily summed by fast Gaussian algorithms. The number of Gaussians can be further reduced via the model reduction based on the balanced truncation based on the square root method. Numerical results on the accuracy and model reduction efficiency show attractive performance of the proposed method.

The magnetic field effects on natural convection of a non-Newtonian power-law nanofluid in the rectangular enclosure have been investigated using the graphics process unit (GPU) accelerated multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). The enclosure is filled up with a power-law non-Newtonian nanofluid with a proper percentage of the nanoparticle volume fraction. The height of the enclosure is twice its width. The left and right walls are heated with constant temperature, and the top and bottom walls are thermally adiabatic. Initially, the code is validated for the Newtonian nanofluid, and then validation is done with non-Newtonian power-law fluids. The numerical results with the effects of magnetic fields are presented in terms of the streamlines, isotherms, temperature distribution, local and average Nusselt number for the shear thinning and thickening nanofluid. The heat transfer rate gets augmented for the shear-thinning fluids $(n<1)$ while that becomes attenuated for the shear-thickening fluids $(n >1).$ Besides, the magnetic field effects reduce the heat transfer rate from the wall to the fluid region.

In this paper the plane wave methods are discussed for solving the time-harmonic elastic wave propagation problems with the complex valued coefficients in two and three space dimensions. The plane wave least-squares method and the ultra-weak variational formulation are developed for the elastic wave propagation. The error estimates of the approximation solutions generated by the PWLS method are derived. Moreover, Combined with local spectral elements, the plane wave methods  are generalized to solve the nonhomogeneous elastic wave problems. Numerical results verify the validity of the theoretical results and indicate that the resulting approximate solution generated by the PWLS method is generally more accurate than that generated by a new variant of the ultra-weak variational formulation method when the Lamé constants $\lambda$ and $\mu$ are complex valued.

In the paper, we are interested in block triangular preconditioning techniques based on algebraic multigrid approach for the large-scale, ill-conditioned and 3-by-3 block-structured systems of linear equations originating from multidimensional three-temperature radiation diffusion equations, discretized in space with the symmetry-preserving finite volume element scheme. Both lower and upper block triangular preconditioners are developed, analyzed theoretically, implemented via the two-level parallelization and tested numerically for such linear systems to demonstrate that they lead to mesh-independent convergence behavior and scale well both algorithmically and in parallel.

We propose the Jacobi spectral Galerkin and Jacobi spectral multi Galerkin methods with their iterated versions for obtaining the superconvergence results of a general class of nonlinear Volterra integral equations with a kernel $x^{\beta}(z-x)^{-\kappa},$ where $0<\kappa<1$, $\beta>0$, which have an Abel-type and an endpoint singularity. The exact solutions for these types of integral equations are singular at the initial point of integration. First, we apply a transformation of independent variables to find a new integral equation with a sufficiently smooth solution. Then we discuss the superconvergence rates for the transformed equation in both uniform and weighted $L^2$-norms. We obtain the order of convergence in Jacobi spectral Galerkin method $\mathcal{O}(N^{\frac{3}{4}-r})$ and $\mathcal{O}(N^{-r})$ in uniform and weighted $L^2$-norms, respectively. Whereas iterated Jacobi spectral Galerkin method converges with the order of convergence $\mathcal{O}(N^{-2r})$ in both uniform and weighted $L^2$-norms. We also show that iterated Jacobi spectral multi Galerkin method converges with the orders $\mathcal{O}(N^{-3r}\log{N})$ and $\mathcal{O}(N^{-3r})$ in uniform and weighted $L^2$-norms, respectively. Theoretical results are verified by numerical illustrations.

A class of high-order compact finite difference schemes combined with a splitting method that preserves accuracy are presented for numerical solutions of the multi-dimensional Burgers' equation. Firstly, the implicit high-order compact difference scheme is used to discretize Burgers' equation by non-linear weights that are required to be calculated at each time stage. Secondly, the sixth-order compact difference scheme in space and the fourth-order Runge-Kutta in time are applied to solve the 1D Burgers' equation. Meanwhile a linear stability analysis indicates the scheme is conditionally stable. Thirdly, the 2D and 3D Burgers' equations are divided into 1D subsystems by the splitting method, then these sub-equations' spatial terms are discretized by the fourth-order compact difference scheme, whereas the time discretizations are unchanged. The analyses of stability and accuracy of the splitting method are given to prove the accuracy of splitting without a significant loss. Finally, the accuracy and reliability of the proposed method are tested by comparing our experimental results with others selected from the available literature. It is shown that the new method has high-resolution properties and can effectively calculate Burgers' equation at large Reynolds number.