• Abstract

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 to 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.

• Abstract

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.

• Abstract

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.

• Abstract

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.

• Abstract

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 Vall\'ee-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.

• Abstract

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.

• Abstract

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 Lam\'{e} constants $\lambda$ and $\mu$ are complex valued.

• Abstract

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.

• Abstract

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.

• Abstract

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 analysises 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.

• Abstract

By using the Feynman-Kac formula and combining with It\^o-Taylor expansion and finite difference approximation, we first develop an explicit third order one-step method for solving decoupled forward backward stochastic differential equations. Then based on the third order one, an explicit fourth order method is further proposed. Several numerical tests are also presented to illustrate the stability and high order accuracy of the proposed methods.

• Abstract

A second order accurate (in time) numerical scheme is analyzed for the slope-selection (SS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. To make the numerical scheme linear while preserving the nonlinear energy stability, we make use of the scalar auxiliary variable (SAV) approach, in which a modified Crank-Nicolson is applied for the surface diffusion part. The energy stability could be derived a modified form, in comparison with the standard Crank-Nicolson approximation to the surface diffusion term. Such an energy stability leads to an $H^2$ bound for the numerical solution. In addition, this $H^2$ bound is not sufficient for the optimal rate convergence analysis, and we establish a uniform-in-time $H^3$ bound for the numerical solution, based on the higher order Sobolev norm estimate, combined with repeated applications of discrete H\"older inequality and nonlinear embeddings in the Fourier pseudo-spectral space. This discrete $H^3$ bound for the numerical solution enables us to derive the optimal rate error estimate for this alternate SAV method. A few numerical experiments are also presented, which confirm the efficiency and accuracy of the proposed scheme.

• Abstract

In this paper, spectral approximations for distributed optimal control problems governed by the Stokes equation are considered. And the constraint set on velocity is stated with $L^2$-norm. Optimality conditions of the continuous and discretized systems are deduced with the Karush-Kuhn-Tucker conditions and a Lagrange multiplier depending on the constraint. To solve the equivalent systems with high accuracy, Galerkin spectral approximations are employed to discretize the constrained optimal control systems. Meanwhile, we adopt a parameter $\lambda$ in the pressure approximation space, which also guarantees the inf-sup condition, and study a priori error estimates for the velocity and pressure. Specially, an efficient algorithm based on the Uzawa algorithm is proposed and its convergence results are investigated with rigorous analyses. Numerical experiments are performed to confirm the theoretical results.

• Abstract

In this paper, several new energy identities of  metamaterial Maxwell's equations with the perfectly electric conducting (PEC) boundary condition are proposed and proved. These new energy identities are  different from the Poynting theorem. By using these new energy identities, it is proved that the Yee scheme on non-uniform rectangular meshes  is stable in the discrete $L^2$ and $H^1$ norms when the Courant-Friedrichs-Lewy (CFL) condition is satisfied. Numerical experiments in two-dimension (2D) and 3D are carried out and confirm our analysis, and the superconvergence in the discrete $H^1$ norm is found.

• Abstract

In this paper, we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations (RSFDEs) with time delay, which constitute an important class of differential equations of practical significance. We develop a novel implicit alternating direction method that can effectively and efficiently tackle the RSFDEs in both two and three dimensions. The numerical method is proved to be uniquely solvable, stable and convergent with second order accuracy in both space and time. Numerical results are presented to verify the accuracy and efficiency of the proposed numerical scheme.

• Abstract

One goal of financial research is to determine fair prices on the financial market. As financial models and the data sets on which they are based are becoming ever larger and thus more complex, financial instruments must be further developed to adapt to the new complexity, with short runtimes and efficient use of memory space. Here we show the effects of combining known strategies and incorporating new ideas to further improve numerical techniques in computational finance.

In this paper we combine an ADI (alternating direction implicit) scheme for the temporal discretization with a sparse grid approach and the combination technique. The later approach considerably reduces the number of "spatial" grid points. The presented standard financial problem for the valuation of American options using the Heston model is chosen to illustrate the advantages of our approach, since it can easily be adapted to other more complex models.

• Abstract

Deformable microcapsules are widely used in industries and also serve as a mechanical model of living biological cells. In this study, we develop a novel method, by integrating a deep convolutional neural network (DCNN) with high-fidelity mechanistic capsule modelling, to identify the membrane constitutive law and estimate associated parameters of a microcapsule from its steady deformed profile in a capillary tube. Compared with conventional inverse methods, the present approach is more accurate and can increase the prediction throughput rate by a few orders of magnitude. It can process capsules with large deformation in inertial flows. Furthermore, the method can predict the capsule membrane shear elasticity, area dilatation modulus and initial inflation from a single steady capsule profile. We explore the mechanism that the DCNN makes decisions by considering its feature maps, and discuss their potential implication on the development of inverse methods. The present method provides a promising tool which may enable high-throughput mechanical characterisation of microcapsules and biological cells in microfluidic flows.

• Abstract

We develop a new conservative Allen-Cahn phase-field model for diblock copolymers in this paper by using the Allen-Cahn type gradient flow approach for the classical Ohta-Kawaski free energy. The change in volume fraction of two composing monomers is eliminated by using a nonlocal Lagrange multiplier. Based on the recently developed stabilized Scalar Auxiliary Variable method, we have further developed an effective numerical scheme to solve the model. The scheme is highly efficient and only two linear and decoupled equations are needed to solve at  every time step. We then prove that the numerical  method is unconditionally energy stable, the stability and accuracy of the new scheme are  demonstrated by numerous numerical examples conducted. By qualitatively comparing the equilibrium solution obtained by the new model and the classic Cahn-Hilliard model, we illustrate the effectiveness of the new model.

• Abstract

In this paper, a high-accuracy numerical scheme is developed for long-time dynamic simulations of 2D and 3D wave propagation phenomena. In the derivation of the present approach, the second order time derivative of the physical quantity in the wave equation is treated as a substitution variable. Based on the temporal discretization with the Krylov deferred correction (KDC) technique, the original wave problem is then converted into the modified Helmholtz equation. The transformed boundary value problem (BVP) in space is efficiently simulated by using the meshless generalized finite difference method (GFDM) with Taylor series after truncating the second and fourth order approximations. The developed scheme is finally verified by four numerical experiments including cases with complicated domains or the temporally piecewise defined source function. Numerical results match with the analytical solutions and results by the COMSOL software, which demonstrates that the developed KDC-GFDM can allow large time-step sizes for wave propagation problems in long-time intervals.

• Abstract

The study on the fingering phenomenon has been playing an important role in understanding the mechanism of the fluid flow through the porous media. In this paper, a numerical method consisting of the Crank-Nicolson scheme for the temporal discretization and the finite element method for the spatial discretization is proposed for the relaxation non-equilibrium Richards equation in simulating the fingering phenomenon. Towards the efficiency and accuracy of the numerical simulations, a predictor-corrector process is used for resolving the nonlinearity of the equation, and an $h$-adaptive mesh method is introduced for accurately resolving the solution around the wetting front region, in which a heuristic a posteriori error indicator is designed for the purpose. In numerical simulations, a traveling wave solution of the governing equation is derived for checking the numerical convergence of the proposed method. The effectiveness of the $h$-adaptive method is also successfully demonstrated by numerical experiments. Finally the mechanism on generating fingers is discussed by numerically studying several examples.