Accurate Numerical Scheme Research Articles

Efficient BBFM-collocation for weakly singular oscillatory Volterra integral equations of the second kind

We present a fast and accurate numerical scheme for approximating weakly singular oscillatory Volterra integral equations (VIEs) of the second kind. The main idea is the combination of the black-box fast multipole method (FMM) and collocation methods that leads to a significant speed-up in CPU time for a given tolerance. The weakly singular oscillatory integrals, which arise from the evaluation of integral equations, are calculated using Clenshaw-Curtis Filon methods. Because the kernel function is approximated by a Chebyshev interpolation scheme, it is very useful for the kernel which is known analytically but is quite complicated. Compared with the method and the error estimates reported previously, the new scheme is suitable for more general kernel function and provide sharper estimates. The efficiency and accuracy of the new method are verified by numerical examples.

A Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Square Phase Field Crystal Equation

In this paper we propose and analyze a second order accurate (in time) numerical scheme for the square phase field crystal equation, a gradient flow modeling crystal dynamics at the atomic scale in space but on diffusive scales in time. Its primary difference with the standard phase field crystal model is an introduction of the 4-Laplacian term in the free energy potential, which in turn leads to a much higher degree of nonlinearity. 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 second order backward differentiation formula is applied in the temporal stencil. Meanwhile, a direct application of the SAV method faces certain difficulties, due to the involvement of the 4-Laplacian term, combined with a derivation of the lower bound of the nonlinear energy functional. In the proposed numerical method, an appropriate decomposition for the physical energy functional is formulated, so that the nonlinear energy part has a well-established global lower bound, and the rest terms lead to constant-coefficient diffusion terms with positive eigenvalues. In turn, the numerical scheme could be very efficiently implemented by constant-coefficient Poisson-like type solvers (via FFT), and energy stability is established by introducing an auxiliary variable, and an optimal rate convergence analysis is provided for the proposed SAV method. A few numerical experiments are also presented, which confirm the efficiency and accuracy of the proposed scheme.

Highly accurate numerical scheme based on polynomial scaling functions for equal width equation

In this paper we established a numerical method for Equal Width (EW) Equation using Polynomial Scaling Functions. The EW equation is a simpler alternative to well known Korteweg de Vries (KdV) and regularized long wave (RLW) equations which have many applications in nonlinear wave phenomena. According to Polynomial scaling method, algebraic polynomials are used to get the orthogonality between the wavelets and corresponding scaling functions with respect to the Chebyshev weight. First we introduce polynomial scaling functions, how are the functions are approximated according to these and Operational matrix of derivatives are given. For time discretization of the function we use finite difference method with Rubin Graves linearization and polynomial scaling functions are used for the space discretization. The method is applied to four different problem and the obtained results are compared with the results in the literature and with the exact results to give the efficiency of the method.

Second-order accurate and energy stable numerical scheme for an immiscible binary mixture of nematic liquid crystals and viscous fluids with strong anchoring potentials

We consider in this paper numerical approximations of the immiscible binary mixture of nematic liquid crystals (LCs) and viscous fluids. We develop a second-order time marching scheme by adopting the recently developed stabilized-SAV (scalar auxiliary variable) approach where several critical stabilization terms are added to enhance the stability; thus, large time steps are allowed in computations. The scheme is highly efficient and one only needs to solve several decoupled linear equations with constant coefficients at each time step. The energy stability of the scheme is proved, and various 2D and 3D numerical experiments including the drop deformations and phase separations are then performed to validate the accuracy and energy stability of the proposed scheme.

Local discontinuous Galerkin approximations to variable-order time-fractional diffusion model based on the Caputo–Fabrizio fractional derivative

In this paper, we construct and investigate an accurate numerical scheme for solving a class of variable-order (VO) fractional diffusion equation based on the Caputo–Fabrizio fractional derivative. The scheme is presented by using a finite difference method in temporal variable and a local discontinuous Galerkin method (LDG) in space. For all variable-order α(t)∈(0,1), we derive the stability and L2 convergence of proposed scheme and prove that the method is of accuracy-order O(τ+hk+1), where τ, h and k are temporal step sizes, spatial step sizes and the degree of piecewise Pk polynomials, respectively. Several numerical tests are given to validate the theoretical analysis and efficiency of the proposed algorithm.

New Kernel Function for the Approximation of Third Order Derivatives

A new kernel function is developed to approximate the third order derivative by mean of the Smoothed Particle Hydrodynamics (SPH) method. It has the advantage to be efficiently used with gridded data and random distributions. Due to the discrepancy of the particles in the former distribution, we extended the use of the CSPM method for the approximation of third order derivatives. Our new kernel function provides three accurate numerical schemes, in conjunction with the trapezoidal rule, Simpson’s rule and the CSPM method respectively.

An efficient stabilized multiple auxiliary variables method for the Cahn–Hilliard–Darcy two-phase flow system

Herein, we develop a totally decoupled, linear, and temporally second-order accurate numerical scheme for the Cahn–Hilliard–Darcy system which models the two-phase incompressible fluid flows in porous medium or in Hele–Shaw cells. Our proposed scheme is based on a simple and efficient stabilized multiple scalar auxiliary variables (S-MSAV) approach. Two time-dependent variables are defined to change the original governing equations to be the equivalent forms and then the Crank–Nicolson type approximation and the explicit Adams–Bashforth approximation are used to temporally discretize the equivalent equations. All nonlinear parts and auxiliary variables are explicitly treated, thus we can decouple the phase-field variable and auxiliary variables in time. The velocity and pressure are decoupled by using a second-order accurate pressure correction method. Therefore, our numerical scheme is very simple and efficient. We analytically prove the discrete energy dissipation law and unique solvability of our scheme with the absence of external force. The benchmark tests are performed to show that our method has desired accuracy and energy stability. Moreover, our method works well in simulating buoyancy-driven pinchoff and viscous fingering.

Uniformly accurate nested Picard integrators for a system of oscillatory ordinary differential equations

A uniformly second order accurate nested Picard iterative integrator (NPI) is proposed to solve a system of oscillatory ordinary equations involving a parameter $$0<\varepsilon \le 1$$ . The solution of this system propagates wave with $$O(\varepsilon ^2)$$ wavelength and $$O(\varepsilon ^4)/O(\varepsilon ^2)$$ amplitude for well-prepared/ill-prepared initial data. Based on the NPI and the exponential integrator, a uniformly accurate (w.r.t $$\varepsilon $$ ) second order numerical scheme with $$O(\tau ^2)$$ error has been developed. The method can be generalized to higher order. Numerical tests are presented to confirm our error estimates.

A generalized optimal fourth-order finite difference scheme for a 2D Helmholtz equation with the perfectly matched layer boundary condition

A crucial part of successful wave propagation related inverse problems is an efficient and accurate numerical scheme for solving the seismic wave equations. In particular, the numerical solution to a multi-dimensional Helmholtz equation can be troublesome when the perfectly matched layer (PML) boundary condition is implemented. In this paper, we present a general approach for constructing fourth-order finite difference schemes for the Helmholtz equation with PML in the two-dimensional domain based on point-weighting strategy. Particularly, we develop two optimal fourth-order finite difference schemes, optimal point-weighting 25p and optimal point-weighting 17p. It is shown that the two schemes are consistent with the Helmholtz equation with PML. Moreover, an error analysis for the numerical approximation of the exact wavenumber is provided. Based on minimizing the numerical dispersion, we implement the refined choice strategy for selecting optimal parameters and present refined point-weighting 25p and refined point-weighting 17p finite difference schemes. Furthermore, three numerical examples are provided to illustrate the accuracy and effectiveness of the new methods in reducing numerical dispersion.

Various choices of source terms for a class of two-fluid two-velocity models

The source terms of the Baer–Nunziato model involve highly non-linear return to equilibrium terms. In order to perform numerical simulations of realistic situations, accounting for this relaxation effects is mandatory. Unfortunately, with the classical forms retained for these source terms in the literature, building efficient, robust and accurate numerical schemes is a tricky task. In this paper, we propose different non-classical forms for these source terms. As for the classical ones, they all agree with the second law of thermodynamics and they are thus associated with a growth of an entropy. The great advantage of some of these new forms of source terms is that they are more linear with respect to the conservative variables. Consequently, this allows to propose more robust, efficient and accurate numerical schemes, in particular when considering fractional step approaches for which source terms and convection terms are solved separately.

Well-balanced finite volume multi-resolution schemes for solving the Ripa models

In this article, fifth order well-balanced finite volume multi-resolution weighted essentially non-oscillatory (FV MR-WENO) schemes are constructed for solving one-dimensional and two-dimensional Ripa models. The Ripa system generalizes the shallow water model by incorporating horizontal temperature gradients. The presence of temperature gradients and source terms in the Ripa models introduce difficulties in developing high order accurate numerical schemes which can preserve exactly the steady-state conditions. The proposed numerical methods are capable to exactly preserve the steady-state solutions and maintain non-oscillatory property near the shock transitions. Moreover, in the procedure of derivation of the FV MR-WENO schemes unequal central spatial stencils are used and linear weights can be chosen any positive numbers with only restriction that their total sum is one. Various numerical test problems are considered to check the validity and accuracy of the derived numerical schemes. Further, the results obtained from considered numerical schemes are compared with those of a high resolution central upwind scheme and available exact solutions of the Ripa model.

On a novel fully-decoupled, linear and second-order accurate numerical scheme for the Cahn–Hilliard–Darcy system of two-phase Hele–Shaw flow

We construct a novel fully-decoupled and second-order accurate time marching numerical scheme with unconditional energy stability for the Cahn–Hilliard–Darcy phase-field model of the two-phase Hele–Shaw flow, in which, the key idea to realize the full decoupling structure is to use the so-called “zero-energy-contribution” function and design a special ordinary differential equation to deal with the nonlinear coupling terms between the flow field and the phase-field variable. Compared with the existing decoupling type schemes, the scheme developed here is more effective, efficient and easy to implement. At each time step, one only needs to solve a few fully-decoupled linear equations only with constant coefficients. We also strictly prove the solvability and unconditional energy stability of the scheme, and implement various numerical simulations in 2D and 3D to show the efficiency and stability of the proposed scheme numerically.

Very high order well-balanced schemes for non-prismatic one-dimensional channels with arbitrary shape

Accurate modelling of shallow water flows in canals for realistic scenarios cannot neglect the variations of the size and shape of the cross-section along the canal. In typical situations, especially in view of applications to flows in networks of canals, a 2D model would be too costly, while the standard 1D Saint-Venant model for a rectangular channel with constant breadth would be too coarse. In this paper we derive efficient, high order accurate and robust numerical schemes for a 1.5D model, in which the canal can have an arbitrary cross-section: the shape can vary along the channel and is described by a depth-dependent breadth function σ(x,z), where x is the coordinate along the channel and z represents the vertical direction. Contrary to all previous schemes for this model, we reformulate the equations in a way that avoids the appearance of the moments of σ and of ∂σ/∂x in the source terms. Our numerical schemes are based on the path-conservative approach for dealing with non-conservative products, are well-balanced on the lake at rest solution and can treat wet-try transitions. They can be implemented with any order of accuracy. Schemes up to third order are explicitly constructed and tested, thanks to the CWENO reconstruction technique. Through a large set of numerical tests we show the performance of the new schemes and compare the results obtained with different orders of accuracy.

Estimation of Rolling Motion of Ship in Random Beam Seas by Efficient Analytical and Numerical Approaches

A steady-state roll motion of ships with nonlinear damping and restoring moments for all times is modeled by a second-order nonlinear differential equation. Analytical expressions for the roll angle, velocity, acceleration, and damping and restoring moments are derived using a modified approach of homotopy perturbation method (HPM). Also, the operational matrix of derivatives of ultraspherical wavelets is used to obtain a numerical solution of the governing equation. Illustrative examples are provided to examine the applicability and accuracy of the proposed methods when compared with a highly accurate numerical scheme.

Energy Stable Arbitrary Order ETD-MS Method for Gradient Flows with Lipschitz Nonlinearity

We present a methodology to construct efficient high-order in time accurate numerical schemes for a class of gradient flows with appropriate Lipschitz continuous nonlinearity. There are several ingredients to the strategy: the exponential time differencing (ETD), the multi-step (MS) methods, the idea of stabilization, and the technique of interpolation. They are synthesized to develop a generic $k^{th}$ order in time efficient linear numerical scheme with the help of an artificial regularization term of the form $A\tau^k\frac{\partial}{\partial t}\mathcal{L}^{p(k)}u$ where $\mathcal{L}$ is the positive definite linear part of the flow, $\tau$ is the uniform time step-size. The exponent $p(k)$ is determined explicitly by the strength of the Lipschitz nonlinear term in relation to $\mathcal{L}$ together with the desired temporal order of accuracy $k$. To validate our theoretical analysis, the thin film epitaxial growth without slope selection model is examined with a fourth-order ETD-MS discretization in time and Fourier pseudo-spectral in space discretization. Our numerical results on convergence and energy stability are in accordance with our theoretical results.

Numerical investigation for the fractional nonlinear space‐time telegraph equation via the trigonometric Quintic B‐spline scheme

This manuscript employs the trigonometric Quintic B‐spline (TQBS) scheme for investigating the numerical solution of the conformable fractional nonlinear time‐space telegraph equation. This model is derived by Oliver Heaviside 1880 to describe the cutting‐edge or voltage of an electrified transmission range with the day yet distance from an electrified transmission or electromagnetic wave's application. In Hamed and Khater, three recent computational schemes (sech–tanh expansion method; extended sinh–Gorden expansion method; and extended simplest equation method) have been applied to the fractional model for constructing the exact traveling wave solutions. Many distinct solutions have been obtained, and some of them have been sketched in two, three‐dimensional, and density plots. Here, these solutions are investigated to evaluate the initial and boundary conditions that apply the suggested numerical scheme. The accuracy of the constructed exact solutions is investigated by calculating the absolute value of error. The most accurate numerical schemes of the three applied computational schemes have been illustrated.

An improved error analysis for a second-order numerical scheme for the Cahn–Hilliard equation

In this paper we present an error analysis for a second order accurate numerical scheme for the 2-D and 3-D Cahn–Hilliard (CH) equation, with an improved convergence constant. The unique solvability, unconditional energy stability, and a uniform-in-time H2 stability of this numerical scheme have already been established. However, a standard error estimate gives a convergence constant in an order of exp(CTε−m0), with m0 a positive integer and the interface width parameter ε being small, which comes from the application of discrete Gronwall inequality. To overcome this well-known difficulty, we apply a spectrum estimate for the linearized Cahn–Hilliard operator (Alikakos and Fusco, 1993; Chen, 1994; Feng and Prohl, 2004), perform a detailed numerical analysis, and get an improved estimate, in which the convergence constant depends on 1ε only in a polynomial order, instead of the exponential order.

A posteriori diffusion analysis of numerical schemes for application to propagating linear waves

AbstractWe compare several higher‐order accurate numerical schemes based on their dissipation characteristics found out in an a posteriori energy analysis. The diffusion analysis is performed based on correlation between the modal energy as well as the total energy of the broadband signal and the dissipation characteristics of the candidate numerical scheme. This technique is used for analyzing linear as well as nonlinear including space–time coupled numerical schemes used in simulation of propagating linear waves, such as in computational electromagnetics and computational aeroacoustics. It is found that the discontinuous Galerkin scheme best preserves the signal energy over the simulation time among all candidate numerical schemes under consideration.

A novel fully-decoupled, second-order and energy stable numerical scheme of the conserved Allen–Cahn type flow-coupled binary surfactant model

In this paper, we establish a binary fluid surfactant model by coupling two mass-conserved Allen–Cahn equations and the Navier–Stokes equations and consider numerical simulations of the developed model. Due to a large number of nonlinear and nonlocal coupling terms in the model, it is very challenging to design an efficient and accurate numerical scheme, especially the full decoupling scheme with second-order time accuracy. We solve this challenge by developing a novel fully-decoupled approach, where the key idea achieving the full decoupling structure is to introduce an ordinary differential equation to deal with the nonlinear coupling terms that satisfy the so-called “zero-energy-contribution” property. In this way, we can easily discretize the coupled nonlinear terms in a fully explicit way, while still maintaining unconditional energy stability. By combining with the projection type method and quadratization approach, at each time step, we only need to solve several fully-decoupled linear elliptic equations with constant coefficients. We strictly prove the solvability of the scheme, prove that the scheme satisfies the unconditional energy stability, and give various 2D and 3D numerical simulations to show its stability and accuracy numerically. As far as the author knows, this is the first fully-decoupled and second-order time-accurate scheme of the flow-coupled phase-field type model.

Investigation of the effects of twinning on the mechanical response of polycrystal magnesium

Hexagonal close-packed (HCP) metals show highly anisotropic mechanical responses due to twinning, slip and the interaction among various slip and twin systems. Each HCP metal shows different sets of activated slip and twin systems. The interaction among slip and twin systems causes highly nonlinear and rapidly changing hardening behaviors in the critical resolved shear stress (CRSS) of each slip/twin system. To accurately understand the anisotropic mechanical response of HCP metals, both single-crystal and polycrystal tests must be performed. The interaction among different slip and twin systems shows different behaviors in the single-crystal and polycrystal settings since the interacting environments are different. The fitting process of the single crystal and polycrystal stress–strain data involves a series of trials and errors to find the correct interaction patterns among various slip and twin systems. The fitting procedures used in previous researches take a lot of times of trials and errors and are not guided by any physical property. Therefore, this study employs a recently proposed new fitting procedure, which is based on a physical property, the saturation strength of the material, to fit magnesium experimental data more efficiently with a less number of trials and errors. Compared to the slip process, twinning is computationally more difficult to take into account due to the directionality of the twin process, i.e., twinning occurs only in one direction, not in the opposite direction unlike the slip process. The rapidly changing highly nonlinear interaction hardening behaviors among various slip and twin systems are computationally challenging. Both of the above computational difficulties require a more robust and accurate numerical scheme than previously used ones to obtain accurate representations of experimental data. Therefore, this study proposes a more accurate and robust numerical scheme for the hardening strengths (CRSS) of twin/slip systems and interaction hardening. The newly proposed scheme is based upon implicit time integration, which enhances accuracy and stability. Using the proposed fitting procedure and the implicit integration scheme for hardening strength (CRSS) and interaction hardening, the experimental stress–strain data of polycrystal magnesium shown in Kelley and Hosford (The plastic deformation of magnesium. Technical report, 1967, Trans Metall Soc AIME 242:5–13, 1968) are successfully reproduced.