Showing papers on "Finite difference published in 2015"
TL;DR: A systematic comparison of six commonly used numerical schemes for 1D’sblood flow modelling, showing a good agreement among all numerical schemes and their ability to capture the main features of pressure, flow and area waveforms in large arteries.
Abstract: Summary
Haemodynamical simulations using one-dimensional (1D) computational models exhibit many of the features of the systemic circulation under normal and diseased conditions. Recent interest in verifying 1D numerical schemes has led to the development of alternative experimental setups and the use of three-dimensional numerical models to acquire data not easily measured in vivo. In most studies to date, only one particular 1D scheme is tested. In this paper, we present a systematic comparison of six commonly used numerical schemes for 1D blood flow modelling: discontinuous Galerkin, locally conservative Galerkin, Galerkin least-squares finite element method, finite volume method, finite difference MacCormack method and a simplified trapezium rule method. Comparisons are made in a series of six benchmark test cases with an increasing degree of complexity. The accuracy of the numerical schemes is assessed by comparison with theoretical results, three-dimensional numerical data in compatible domains with distensible walls or experimental data in a network of silicone tubes. Results show a good agreement among all numerical schemes and their ability to capture the main features of pressure, flow and area waveforms in large arteries. All the information used in this study, including the input data for all benchmark cases, experimental data where available and numerical solutions for each scheme, is made publicly available online, providing a comprehensive reference data set to support the development of 1D models and numerical schemes. Copyright © 2015 John Wiley & Sons, Ltd.
163 citations
TL;DR: The objective of this paper is to establish that RBF–PUM is viable for parabolic PDEs of convection–diffusion type and it is shown that using a node layout that captures the solution features improves the accuracy significantly compared with a uniform node distribution.
Abstract: Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations (PDEs) because they are flexible with respect to geometry, they can provide high order convergence, they allow for local refinement, and they are easy to implement in higher dimensions. For global RBF methods, one of the major disadvantages is the computational cost associated with the dense linear systems that arise. Therefore, research is currently directed towards localized RBF approximations such as the RBF partition of unity collocation method (RBF---PUM) proposed here. The objective of this paper is to establish that RBF---PUM is viable for parabolic PDEs of convection---diffusion type. The stability and accuracy of RBF---PUM is investigated partly theoretically and partly numerically. Numerical experiments show that high-order algebraic convergence can be achieved for convection---diffusion problems. Numerical comparisons with finite difference and pseudospectral methods have been performed, showing that RBF---PUM is competitive with respect to accuracy, and in some cases also with respect to computational time. As an application, RBF---PUM is employed for a two-dimensional American option pricing problem. It is shown that using a node layout that captures the solution features improves the accuracy significantly compared with a uniform node distribution.
152 citations
TL;DR: In this paper, a numerical solution of MHD flow of tangent hyperbolic fluid model over a stretching cylinder is obtained, where the governing boundary layer equation is converted into an ordinary differential equation using similarity transformations, which is then solved numerically by applying the implicit finite difference Keller box method.
139 citations
TL;DR: In this article, the stability area for the difference system with the Caputo forward difference operator is described. But the stability conditions are not explicit and involve the decay rate of the solutions.
Abstract: Abstract The paper describes the stability area for the difference system (Δαy)(n + 1 − α) = Ay(n), n= 0, 1, . . . , with the Caputo forward difference operator Δα of a real order α ∈ (0, 1) and a real constant matrix A. Contrary to the existing result on this topic, our stability conditions are fully explicit and involve the decay rate of the solutions. Some comparisons with a difference system of the Riemann- Liouville type are discussed as well, including related consequences and illustrating examples.
131 citations
TL;DR: A finite difference/finite element algorithm, which is based on a finite difference approximation in time direction and finite element method in spatial direction, is presented and discussed to cast about for the numerical solutions of a time-fractional fourth-order reaction–diffusion problem with a nonlinear reaction term.
Abstract: In this article, a finite difference/finite element algorithm, which is based on a finite difference approximation in time direction and finite element method in spatial direction, is presented and discussed to cast about for the numerical solutions of a time-fractional fourth-order reaction–diffusion problem with a nonlinear reaction term. To avoid the use of higher-order elements, the original problem with spatial fourth-order derivative need to be changed into a second-order coupled system by introducing an intermediate variable σ = Δ u . Then the fully discrete finite element scheme is formulated by using a finite difference approximation for time fractional and integer derivatives and finite element method in spatial direction. The unconditionally stable result in the norm, which just depends on initial value and source item, is derived. Some a priori estimates of L 2 -norm with optimal order of convergence O ( Δ t 2 − α + h m + 1 ) , where Δ t and h are time step length and space mesh parameter, respectively, are obtained. To confirm the theoretical analysis, some numerical results are provided by our method.
129 citations
TL;DR: In this paper, a method based on radial basis function (RBF)-generated finite differences (FD) was proposed for numerically solving diffusion and reaction diffusion equations (PDEs) on closed surfaces embedded in the Euclidean plane.
Abstract: In this paper, we present a method based on radial basis function (RBF)-generated finite differences (FD) for numerically solving diffusion and reaction---diffusion equations (PDEs) on closed surfaces embedded in $${\mathbb {R}}^d$$Rd. Our method uses a method-of-lines formulation, in which surface derivatives that appear in the PDEs are approximated locally using RBF interpolation. The method requires only scattered nodes representing the surface and normal vectors at those scattered nodes. All computations use only extrinsic coordinates, thereby avoiding coordinate distortions and singularities. We also present an optimization procedure that allows for the stabilization of the discrete differential operators generated by our RBF-FD method by selecting shape parameters for each stencil that correspond to a global target condition number. We show the convergence of our method on two surfaces for different stencil sizes, and present applications to nonlinear PDEs simulated both on implicit/parametric surfaces and more general surfaces represented by point clouds.
120 citations
TL;DR: The superconvergent points of the traditional first-order Grunwald-Letnikov formula to approximate the Riemann-Liouville fractional derivatives are used to construct the several finite difference schemes for solving a class of time-fractional subdiffusion equations.
104 citations
TL;DR: A new class of two-dimensional fully nonlinear and weakly dispersive Green-Naghdi equations over varying topography that have a mathematical structure which makes them much more suitable for the numerical resolution, in particular in the demanding case of two dimensional surfaces.
102 citations
TL;DR: This paper rigorously prove first order convergence in time and second or- der convergence in space, and establishes a finite difference analog of a Gagliardo-Nirenberg type inequality.
Abstract: We present an error analysis for an unconditionally energy stable, fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equa- tion, a modified Cahn-Hilliard equation coupled with the Darcy flow law. The scheme, proposed by S. M. Wise, is based on the idea of convex splitting. In this paper, we rigorously prove first order convergence in time and second or- der convergence in space. Instead of the (discrete)L ∞ (0,T;L 2 )∩L 2 (0,T;H 2 h ) error estimate, which would represent the typical approach, we provide a dis- creteL ∞ (0,T;H 1 )∩L 2 (0,T;H 3 h )e rror estimate for the phase variable, which allows us to treat the nonlinear convection term in a straightforward way. Our convergence is unconditionalin the sense that the time stepsis in no way constrained by the mesh spacingh .T his is accomplished with the help of anL 2 (0,T;H 3 h ) bound of the numerical approximation of the phase variable. To facilitate both the stability and convergence analyses, we establish a finite difference analog of a Gagliardo-Nirenberg type inequality.
94 citations
TL;DR: The aim of this paper is to show that the meshless method based on the weak form is also suitable for the treatment of the nonlinear partial differential equations and to obtain an error bound for the new method.
91 citations
TL;DR: In this work, a loosely coupled scheme for Stokes–Biot equations is developed and thoroughly analyzed, based on Nitsche’s method for enforcing interface conditions, and the application of the loosely coupling scheme as a preconditioner for the monolithic approach is considered.
TL;DR: In this article, the authors compared three classes of higher-order shock capturing schemes: (1) central finite-difference schemes with explicit artificial dissipation, (2) a compact centered finite difference scheme with localized artificial diffusivity and (3) weighted essentially nonoscillatory schemes in both explicit and compact finite difference forms.
TL;DR: Several one- and two-dimensional numerical examples are used to demonstrate accuracy, robustness, and effectiveness of the proposed physical-constraints-preserving schemes in solving RHD problems with large Lorentz factor, or strong discontinuities, or low rest-mass density or pressure etc.
TL;DR: A fast and stable high order numerical method for solving Cahn–Hilliard equations for long-time integrations of microstructure coarsening is developed.
TL;DR: In this article, the free vibration of a fractional viscoelastic Timoshenko nanobeam is studied through inserting fractional calculus as a viscocelastic material compatibility equations in nonlocal beam theory.
Abstract: In this article, the free vibration of a fractional viscoelastic Timoshenko nanobeam is studied through inserting fractional calculus as a viscoelastic material compatibility equations in nonlocal beam theory. The material properties of a single-walled carbon nanotube (SWCNT) are used and two solution procedures are proposed to solve the obtained equations in the time domain. The former is a semi-analytical approach in which the Galerkin scheme is employed to discretize the governing equations in the spatial domain and the obtained set of ordinary differential equations is solved using a direct numerical integration scheme. On the contrary, the latter is entirely numerical in which the governing equations of system on the spatial and time domains are first discretized using general differential quadrature (GDQ) technique and finite difference (FD) scheme, respectively and then the set of algebraic equations is solved to arrive at the time response of system under different boundary conditions. Considering the second solution procedure as the main approach, its validity and accuracy are verified by the semi-analytical approach which is more difficult to enter various boundary conditions. Numerical results are also presented to get an insight into the effects of fractional derivative order, nonlocal parameter, viscoelasticity coefficient and nanobeam length on the time response of fractional viscoelastic Timoshenko nanobeams under different boundary conditions.
TL;DR: In this article, the numerical solutions of one dimensional modified Burgers' equation with the help of Haar wavelet method are investigated, and the calculated numerical solutions are drawn graphically.
Abstract: In this paper, we investigate the numerical solutions of one dimensional modified Burgers’ equation with the help of Haar wavelet method. In the solution process, the time derivative is discretized by finite difference, the nonlinear term is linearized by a linearization technique and the spatial discretization is made by Haar wavelets. The proposed method has been tested by three test problems. The obtained numerical results are compared with the exact ones and those already exist in the literature. Also, the calculated numerical solutions are drawn graphically. Computer simulations show that the presented method is computationally cheap, fast, reliable and quite good even in the case of small number of grid points.
TL;DR: In this article, a numerical method to deal with the nonlinear time-fractional integro-differential reaction-diffusion equation defined by the Caputo fractional derivative is proposed.
Abstract: This paper proposes a numerical method to deal with the nonlinear time-fractional integro-differential reaction-diffusion equation defined by the Caputo fractional derivative. In the proposed method, the space variable is eliminated by using finite difference θ-method to enjoy the stability condition. The method benefits from the radial basis function collocation method, in which the generalized thin plate splines (GTPS) radial basis functions are used. Therefore, it does not require any struggle to determine the shape parameter. The obtained results for some numerical examples reveal that the proposed technique is very effective, convenient and quite accurate to such considered problems.
TL;DR: For 2D and 3D frequency-domain elastic wave modeling, a parallel iterative solver based on a conjugate gradient acceleration of the symmetric Kaczmarz row-projection method, named the conjugates-gradient-accelerated component-averaged row projections (CARP-CG) method, shows interesting convergence properties.
Abstract: Full-waveform inversion and reverse time migration rely on an efficient forward-modeling approach. Current 3D large-scale frequency-domain implementations of these techniques mostly extract the desired frequency component from the time-domain wavefields through discrete Fourier transform. However, instead of conducting the time-marching steps for each seismic source, in which the time step is limited by the stability condition, performing the wave modeling directly in the frequency domain using an iterative linear solver may reduce the entire computational complexity. For 2D and 3D frequency-domain elastic wave modeling, a parallel iterative solver based on a conjugate gradient acceleration of the symmetric Kaczmarz row-projection method, named the conjugate-gradient-accelerated component-averaged row projections (CARP-CG) method, shows interesting convergence properties. The parallelization is realized through row-block division and component averaging operations. Convergence is achieved systemat...
TL;DR: In this article, an approximation method for advection-diffusion-reaction equa-tions where the (generalized) degrees of freedom are polynomials of order k>=0 at mesh faces is presented.
Abstract: We design and analyze an approximation method for advection-diffusion-reaction equa-tions where the (generalized) degrees of freedom are polynomials of order k>=0 at mesh faces. The method hinges on local discrete reconstruction operators for the diffusive and advective derivatives and a weak enforcement of boundary conditions. Fairly general meshes with poly-topal and nonmatching cells are supported. Arbitrary polynomial orders can be considered, including the case k=0 which is closely related to Mimetic Finite Difference/Mixed-Hybrid Finite Volume methods. The error analysis covers the full range of Peclet numbers, including the delicate case of local degeneracy where diffusion vanishes on a strict subset of the domain. Computational costs remain moderate since the use of face unknowns leads to a compact stencil with reduced communications. Numerical results are presented.
TL;DR: The present method is applied to three different fluid-structure interaction problems: flows around a flexible filament, a flapping flag in a free stream, and a flexible flapping wing in normal hovering, respectively.
TL;DR: The proposed interface coupling technique has been successfully combined with a high order entropy stable discretization for the simulation of two-dimensional and 3D viscous subsonic and supersonic flows and provides a simple and automatic way to compute the magnitude of the viscous IP term.
TL;DR: In this article, a numerical optimisation procedure is presented to obtain non-linear elastic viscoplastic (EVP) model parameters adopting the available consolidation data, where the Crank-Nicolson finite difference scheme is applied to solve the combination of coupled partial differential equations of the EVP model and the consolidation theory.
TL;DR: In this article, a semi-implicit space-time discontinuous Galerkin finite element (SDFE) method was proposed for the solution of the two dimensional incompressible Navier-Stokes equations on staggered unstructured triangular meshes, where isoparametric finite elements are used to take into account curved domain boundaries.
TL;DR: In this article, the authors developed novel optimal time-space-domain finite-difference (SFD) schemes for acoustic and elastic-wave-equation modeling, where a fourth-order multiextreme value objective function with respect to FD coefficients was involved.
Abstract: Staggered-grid finite-difference (SFD) methods are widely used in modeling seismic-wave propagation, and the coefficients of finite-difference (FD) operators can be estimated by minimizing dispersion errors using Taylor-series expansion (TE) or optimization. We developed novel optimal time-space-domain SFD schemes for acoustic- and elastic-wave-equation modeling. In our schemes, a fourth-order multiextreme value objective function with respect to FD coefficients was involved. To yield the globally optimal solution with low computational cost, we first used variable substitution to turn our optimization problem into a quadratic convex one and then used least-squares (LS) to derive the optimal SFD coefficients by minimizing the relative error of time-space-domain dispersion relations over a given frequency range. To ensure the robustness of our schemes, a constraint condition was imposed that the dispersion error at each frequency point did not exceed a given threshold. Moreover, the hybrid absorbin...
TL;DR: In this paper, two stable and one conditionally stable finite difference schemes of second-order in both time and space for the time-fractional diffusion-wave equation were proposed.
Abstract: We propose two stable and one conditionally stable finite difference schemes of second-order in both time and space for the time-fractional diffusion-wave equation. In the first scheme, we apply the fractional trapezoidal rule in time and the central difference in space. We use the generalized Newton---Gregory formula in time for the second scheme and its modification for the third scheme. While the second scheme is conditionally stable, the first and the third schemes are stable. We apply the methodology to the considered equation with also linear advection---reaction terms and also obtain second-order schemes both in time and space. Numerical examples with comparisons among the proposed schemes and the existing ones verify the theoretical analysis and show that the present schemes exhibit better performances than the known ones.
TL;DR: In this article, a new technique for a finite-difference weighted essentially nonoscillatory scheme (WENO) on curvilinear grids to preserve freestream is introduced.
TL;DR: This paper utilizes the maximum-principle-preserving flux limiting technique, originally designed for high-order weighted essentially nonoscillatory (WENO) methods for scalar hyperbolic conservation laws, to develop a class of high- order positivity- Preserving finite difference WENO methods for the ideal magnetohydrodynamic equations.
Abstract: In this paper, we utilize the maximum-principle-preserving flux limiting technique, originally designed for high-order weighted essentially nonoscillatory (WENO) methods for scalar hyperbolic conservation laws, to develop a class of high-order positivity-preserving finite difference WENO methods for the ideal magnetohydrodynamic equations. Our scheme, under the constrained transport framework, can achieve high-order accuracy, a discrete divergence-free condition, and positivity of the numerical solution simultaneously. Numerical examples in one, two, and three dimensions are provided to demonstrate the performance of the proposed method.
TL;DR: It is shown that temperatures and pressures may be significantly overestimated in calculations of shock-induced bubble collapse in water if temperature errors are not prevented, and a three-dimensional, high-order accurate, solution-adaptive finite difference framework is developed.
TL;DR: In this paper, the authors analyze the accuracy of several monolithic methods for wave propagation in the presence of a fluid-solid interface and demonstrate that a first-order velocity-stress formulation can, in general, be used in dealing with fluid solid interfaces without using staggered grids necessarily.
Abstract: The numerical simulation of wave propagation in media with solid and fluid layers is essential for marine seismic exploration data analysis. The numerical methods for wave propagation that are applicable to this physical settings can be broadly classified as partitioned or monolithic: The partitioned methods use separate simulations in the fluid and solid regions and explicitly satisfy the interface conditions, whereas the monolithic methods use the same method in all the domain without any special treatment of the fluid-solid interface. Despite the accuracy of the partitioned methods, the monolithic methods are more common in practice because of their convenience. In this paper, we analyse the accuracy of several monolithic methods for wave propagation in the presence of a fluid-solid interface. The analysis is based on grid-dispersion criteria and numerical examples. The methods studied here include: the classical finite-difference method (FDM) based on the second-order displacement formulation of the elastic wave equation (DFDM), the staggered-grid finite difference method (SGFDM), the velocity-stress FDM with a standard grid (VSFDM) and the spectral-element method (SEM). We observe that among these, DFDM and the first-order SEM have a large amount of grid dispersion in the fluid region which renders them impractical for this application. On the other hand, SGFDM, VSFDM and SEM of order greater or equal to 2 yield accurate results for the body waves in the fluid and solid regions if a sufficient number of nodes per wavelength is used. All of the considered methods yield limited accuracy for the surface waves because the proper boundary conditions are not incorporated into the numerical scheme. Overall, we demonstrate both by analytic treatment and numerical experiments, that a first-order velocity-stress formulation can, in general, be used in dealing with fluid-solid interfaces without using staggered grids necessarily.
TL;DR: In this article, a series of numerical simulations compared with laboratory dam-break experiments were carried out, which demonstrates that the current computational framework is able to determine the dam break hydraulics over erodible sediment.
Abstract: Coupled shallow water equations integrated with sediment transport and morphological evolution are presented in this paper. The momentum exchange terms that originated from the interaction between flow and sediment, which were ignored by several researchers, are taken into account. The time and space second-order, MacCormack total variation diminishing (TVD) finite difference method is used to solve these equations. A series of numerical simulations compared with laboratory dam-break experiments were carried out. The simulated results are in good agreement with experimental measured results, which demonstrates that the current computational framework is able to determine the dam-break hydraulics over erodible sediment.