Showing papers on "Temporal discretization published in 2009"
TL;DR: In this article, the authors identify a crucial numerical problem in sigma coordinate models, leading to unacceptable spurious diapycnal mixing and propose RSUP3, which is a modified version of the third-order up-wind advection scheme.
157 citations
TL;DR: In this article, the authors proposed a global-local split of the system of equations in which the fast action potential is introduced as a nodal degree of freedom, whereas the local variable is the slow recovery variable introduced as an internal variable on the integration point level.
Abstract: The key objective of this work is the design of an unconditionally stable, robust, efficient, modular, and easily expandable finite element-based simulation tool for cardiac electrophysiology. In contrast to existing formulations, we propose a global–local split of the system of equations in which the global variable is the fast action potential that is introduced as a nodal degree of freedom, whereas the local variable is the slow recovery variable introduced as an internal variable on the integration point level. Cell-specific excitation characteristics are thus strictly local and only affect the constitutive level. We illustrate the modular character of the model in terms of the FitzHugh–Nagumo model for oscillatory pacemaker cells and the Aliev–Panfilov model for non-oscillatory ventricular muscle cells. We apply an implicit Euler backward finite difference scheme for the temporal discretization and a finite element scheme for the spatial discretization. The resulting non-linear system of equations is solved with an incremental iterative Newton–Raphson solution procedure. Since this framework only introduces one single scalar-valued variable on the node level, it is extremely efficient, remarkably stable, and highly robust. The features of the general framework will be demonstrated by selected benchmark problems for cardiac physiology and a two-dimensional patient-specific cardiac excitation problem. Copyright © 2009 John Wiley & Sons, Ltd.
94 citations
TL;DR: In this paper, a solution procedure based on the 3D elasticity theory is presented to accurately determine the dynamic response of cross-ply laminated thick plates subjected to moving load, and the results are compared with those of the exact solutions obtained for the isotropic plates under moving load and other available solutions.
75 citations
TL;DR: In this paper, an approach for simulating a precipitation process which is described by a population balance system consisting of the incompressible Navier-Stokes equations, nonlinear convection-diffusion-reaction equations and a transport equation for the particle size distribution (PSD) is presented.
73 citations
TL;DR: This paper formulates and analyzes fully discrete schemes for the two-dimensional Keller-Segel chemotaxis model and considers Cartesian grids and proves fully discrete error estimates for the proposed methods.
Abstract: This paper formulates and analyzes fully discrete schemes for the two-dimensional Keller-Segel chemotaxis model. The spatial discretization of the model is based on the discontinuous Galerkin methods and the temporal discretization is based either on Forward Euler or the second order explicit total variation diminishing (TVD) Runge-Kutta methods. We consider Cartesian grids and prove fully discrete error estimates for the proposed methods. Our proof is valid for pre-blow-up times since we assume boundedness of the exact solution.
52 citations
TL;DR: A robust numerical model with efficient advanced approximations for both spatial and temporal discretization of the coupled flow-transport system in order to reduce the excessive computational requirement while maintaining accuracy.
50 citations
TL;DR: In this article, a mesh-free method named element-free characteristic Galerkin method (EFCGM) is proposed for solving Burgers' equation with various values of viscosity.
Abstract: A new meshfree method named element-free characteristic Galerkin method (EFCGM) is proposed for solving Burgers’ equation with various values of viscosity. Based on the characteristic method, the convection terms of Burgers’ equation disappear and this process makes Burgers’ equation self-adjoint, which ensures that the spatial discretization by the Galerkin method can be optimal. After the temporal discretization, element-free Galerkin method is then applied to solve the semi-discrete equation in space. Moreover, the process is fully explicit at each time step. In order to show the efficiency of the presented method, one-dimensional and two-dimensional Burgers’ equations are considered. The numerical solutions obtained with different values of viscosity are compared with the analytical solutions as well as the results by other numerical schemes. It can be easily seen that the proposed method is efficient, robust and reliable for solving Burgers’ equation, even involving high Reynolds number for which the analytical solution fails.
50 citations
01 Sep 2009
TL;DR: A mesh-free computational method based on radial basis functions in a finite difference mode (RBF-FD) has been developed for the incompressible Navier—Stokes equations in stream function vorticity form, and hence advancing solutions will in time be of comparatively lower cost.
Abstract: Meshless methods for solving fluid and fluid-structure problems have become a promising alternative to the finite volume and finite element methods. In this paper, a mesh-free computational method based on radial basis functions in a finite difference mode (RBF-FD) has been developed for the incompressible Navier—Stokes (NS) equations in stream function vorticity form. This compact RBF-FD formulation generates sparse coefficient matrices, and hence advancing solutions will in time be of comparatively lower cost. The spatial discretization of the incompressible NS equations is done using the RBF-FD method and the temporal discretization is achieved by explicit Euler time-stepping and the Crank—Nicholson method. A novel ghost node strategy is used to incorporate the no-slip boundary conditions. The performance of the RBF-FD scheme with the ghost node strategy is validated against a variety of benchmark problems, including a model fluid—structure interaction problem, and is found to be in a good agreement with the existing results. In addition, a higher-order RBF-FD scheme (which uses ideas from Hermite interpolation) is then proposed for solving the NS equations.
48 citations
TL;DR: In this article, an immersed-boundary technique which is adopted from the direct-forcing method is added to the Navier-Stokes equations to solve fluid-solid interactions, and the second-order Adam-Bashford scheme is used for the temporal discretization while the diffusive and convective terms are discretized using the secondorder central difference and upwind schemes, respectively.
Abstract: We describe an immersed-boundary technique which is adopted from the direct-forcing method. A virtual force based on the rate of momentum changes of a solid body is added to the Navier–Stokes equations. The projection method is used to solve the Navier–Stokes equations. The second-order Adam–Bashford scheme is used for the temporal discretization while the diffusive and the convective terms are discretized using the second-order central difference and upwind schemes, respectively. Some benchmark problems for both stationary and moving solid object have been simulated to demonstrate the capability of the current method in handling fluid–solid interactions.
47 citations
TL;DR: It is shown that full preconditioning is beneficial for explicit time integration while the implicit scheme turns out to be efficient and accurate using just the modified numerical flux function.
41 citations
TL;DR: A series of laboratory-scale 2D tank experiments were conducted and accurately simulated for density driven flow problems on homogeneous porous media to enable a detailed testing of numerical variable-density flow codes under heterogeneous conditions.
TL;DR: In this article, it was shown that the secondary instability occurs in a system with line-tied magnetic fields and boundary shearing, leading to a transition to turbulence which leads to enhanced dissipation of magnetic and kinetic energy.
Abstract: The observed energy-loss rate from the solar corona implies that the coronal magnetic field has a critical angle at which energy is released. It has been hypothesized that at this critical angle an "explosive instability" would occur, leading to an enhanced conversion of magnetic energy into heat. In earlier investigations, we have shown that a shear-dependent magnetohydrodynamic process called "secondary instability" has many of the distinctive features of the hypothetical "explosive instability." In this paper, we give the first demonstration that this "secondary instability" occurs in a system with line-tied magnetic fields and boundary shearing—basically the situation described by Parker. We also show that, as the disturbance due to secondary instability attains finite amplitude, there is a transition to turbulence which leads to enhanced dissipation of magnetic and kinetic energy. These results are obtained from numerical simulations performed with a new parallelized, viscoresistive, three-dimensional code that solves the cold plasma equations. The code employs a Fourier collocation—finite difference spatial discretization, and uses a third-order Runge-Kutta temporal discretization.
TL;DR: In this paper, a time-domain boundary element method (BEM) in conjunction with a multi-domain technique is developed for the crack analysis in two-dimensional (2D), layered, anisotropic and linear elastic solids.
Abstract: Transient elastodynamic crack analysis in two-dimensional (2D), layered, anisotropic and linear elastic solids is presented in this paper. A time-domain boundary element method (BEM) in conjunction with a multi-domain technique is developed for this purpose. Time-domain elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids are applied in the present time-domain BEM. The spatial discretization of the boundary integral equations is performed by a Galerkin-method, while a collocation method is adopted for the temporal discretization of the arising convolution integrals. An explicit time-stepping scheme is developed to compute the unknown boundary data and the crack-opening-displacements (CODs). To show the effects of the crack configuration, the material anisotropy, the layer combination and the dynamic loading on the dynamic stress intensity factors and the scattered elastic wave fields, several numerical examples are presented and discussed.
TL;DR: This work applied a second-order conservative nonlinear multigrid method for the ternary Cahn-Hilliard system with a concentration dependent degenerate mobility for a model for phase separation in a Ternary mixture and proved stability of the numerical solution for a sufficiently small time step.
TL;DR: First compatible temporal interpolations with conveniently usable time integrators are introduced based on stability analysis of the delay differential equations (DDE) and the orthogonal weighted Laguerre polynomials are set forth to provide unconditionally stable schemes.
TL;DR: A high-order finite volume method is employed to solve the local energy approximation model equations for a radio-frequency plasma discharge in a one-dimensional geometry and the so called deferred correction technique, along with high- order Lagrange polynomials is used to calculate the convection and diffusion fluxes.
03 Feb 2009
TL;DR: The symplecticity of this method is proved, which is an expected behavior of the molecular dynamics integrators, and the relevance of this algorithm is numerically demonstrated by applying it to three-dimensional atomic lattices on parallel computer architectures.
Abstract: In this paper we propose a new parareal algorithm for parallelizing in time molecular dynamics problems. The original structure of this algorithm allows one to consider multi-time stepping, namely two levels of temporal discretization, providing a larger range for the fine and coarse solvers definition. We also prove the symplecticity of this method, which is an expected behavior of the molecular dynamics integrators. The relevance of this algorithm is numerically demonstrated by applying it to three-dimensional atomic lattices on parallel computer architectures. For lattices of more than 20000 atoms we get attractive speed-up with proper choice for the coarse solver definition and for the number of processors.
TL;DR: The symplectic integrator for the numerical solution of a kind of high order Schrodinger equation with trapped terms is discussed, and it is proved that it preserves the charge of the original equation veraciously.
TL;DR: A discontinuous Galerkin method designed to improve the accuracy and efficiency of the steady‐state solution at very low Mach number flows using an explicit scheme based on a perturbed formulation of the compressible Euler equations.
Abstract: In this work we present a Discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of the solution at very low Mach number flows using an explicit scheme for the temporal discretization of the compressible Euler equations. The algorithm is based on a perturbed formulation of the governing equ
ations and employs the preconditioning of both the instationary term of the governing equations and the dissipative term of the numerical flux function (full preconditioning approach). The performance of the method is demonstrated by solving an inviscid flow past a NACA0012 airfoil at different very low Mach numbers using various degrees of polynomial approximations. We present numerical results computed with and without perturbed variables which illustrate the influence of the cancellation errors on both the convergence and the accuracy of the DG solutions at low Mach numbers.
TL;DR: A hypersingular time-domain boundary element method (BEM) for transient elastodynamic crack analysis in two-dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this article.
Abstract: A hypersingular time-domain boundary element method (BEM) for transient elastodynamic crack analysis in two-dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack-faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time-stepping scheme is obtained to compute the unknown boundary data including the crack-opening-displacements (CODs). Special crack-tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time-domain BEM. Copyright © 2008 John Wiley & Sons, Ltd.
05 Jan 2009
TL;DR: The numerical behavior of two implicit time-marching methods is investigated in solving two-dimensional unsteady compressible flows, and the Newton-Krylov method is seen to be an efficient choice for implicittime-accurate computations.
Abstract: The numerical behavior of two implicit time-marching methods is investigated in solving two-dimensional unsteady compressible flows. The two methods are the second-order multistep backward differencing formula and the fourth-order multistage explicit first stage, single-diagonal coefficient, diagonally implicit Runge-Kutta scheme. A Newton-Krylov method is used to solve the nonlinear problem arising from the implicit temporal discretization. The methods are studied for two test cases: laminar flow over a cylinder and turbulent flow over a NACA0012 airfoil with a blunt trailing edge. Parameter studies show that the subiteration termination criterion plays a major role in the efficiency of time-marching methods. Efficiency studies show that when only modest global accuracy is needed, the second-order method is preferred. The fourth-order method is more efficient when high accuracy is required. The Newton-Krylov method is seen to be an efficient choice for implicit time-accurate computations.
TL;DR: In this article, a time-domain boundary element method (BEM) together with the sub-domain technique is applied to study dynamic interfacial crack problems in two-dimensional (2D), piecewise homogeneous, anisotropic and linear elastic bi-materials.
TL;DR: In this article, the authors compare the simulation results of two conceptual water balance models using different spatio-temporal discretizations and conclude that the use of monthly data is not inferior to the usage of daily data.
Abstract: To determine the distribution of water balance components in space and time, models are applied with a wide range of spatio-temporal discretizations—from lumped to distributed in the spatial scale and from annual to daily (or shorter) time-steps in the temporal scale. We present a comparative case study where we compare the simulation results of two conceptual water balance models using different spatio-temporal discretizations. Such a comparison enables to assess if different models with different discretizations may still yield similar results in space and time. The study focuses on the mountainous catchment of the river Gail (app. 1300 km2) in southern Austria for the period 1971–1990. The first model uses a semi-distributed discretization and daily data, whereas the second model uses a spatially distributed discretization (1 × 1 km raster) and monthly data. Both models use precipitation and temperature data as input. Parameters of the daily model were calibrated with runoff data of several gauges as part of a study focusing specifically on the Gail catchment. The distributed parameters of the monthly model were estimated regionally for establishing the water balance of the Hydrological Atlas of Austria. Both models perform equally well for runoff simulations. For simulation of temporal dynamics the models agree well for the main inputs and outputs of the system, with slightly lower agreements for sub-components—such as snowmelt for instance. In the spatial domain the correlation between the models is significantly lower. Differences are mainly related to different calibration approaches and are not dependent on the spatio-temporal discretization. Overall, the two water balance models yield consistent results, suggesting that the usage of monthly data is not inferior to the usage of daily data. Copyright © 2008 John Wiley & Sons, Ltd.
TL;DR: In this paper, a general family of Galerkin methods for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains was constructed.
Abstract: We construct a general family of Galerkin methods for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain Ω in R d, d=2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function which satisfies a Fokker-Planck type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. We focus on finitely-extensible nonlinear elastic, FENE-type, dumbbell models. In the case of a corotational drag term we perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that a (sub)sequence of numerical solutions converges to a weak solution of this coupled Navier-Stokes-Fokker-Planck system.
TL;DR: An approximate method is developed herein for coordinate mapping of 2D (vertically-averaged) transport solutions to non-uniform steady-state irrotational and divergence-free flow fields in single-layer aquifers.
TL;DR: In this paper, a comparison of two hypersingular time-domain boundary element methods for transient dynamic crack analysis in two-dimensional, homogeneous, anisotropic and linear elastic solids is presented.
TL;DR: Gradient reproducing kernel particle method is a meshless technique which incorporates the first gradients of the function into the reproducing equation of RKPM, and consideration of non-uniform distribution of particles demonstrates the virtue of the presented methodology in producing smooth profiles in the critical regions near the fronts.
Abstract: Summary Gradient reproducing kernel particle method (GRKPM) is a meshless technique which incorporates the first gradients of the function into the reproducing equation of RKPM. Therefore, in two-dimensionalspace GRKPM introducesthree types of shape functions rather than one. The robustness of GRKPM’s shape functions is established by reconstruction of a third-order polynomial. To enforce the essential boundary conditions (EBCs), GRKPM’s shape functions are modified by transformation technique. By utilizing the modified shape functions, the weak form of the nonlinear evolutionary Buckley-Leverett (BL) equation is discretized in space, rendering a system of nonlinear ordinary differential equations (ODEs). Subsequently, Gear’s method is applied for temporal discretization of the ODEs. Through numerical experiments, employment of a moderate viscosity seeks the efficacy ofthesolutionwhen thediffusionterm isimportant;moreover, applicationof a small viscosity confirms the potential of the approach for treatment of the problems involving steep gradient regions. The outcomes are verified by performing convergence tests using uniformly spaced particles. Consideration of non-uniform distribution of particles further demonstrates the virtue of the presented methodology in producing smooth profiles in the critical regions near the fronts.
05 Jan 2009
TL;DR: In this article, a high-order implicit discontinuous Galerkin method is developed for the time-accurate solutions to the compressible Navier-Stokes equations, where polynomial solutions are represented using a Taylor basis.
Abstract: A high-order implicit discontinuous Galerkin method is developed for the time-accurate solutions to the compressible Navier–Stokes equations. The spatial discretization is carried out using a high order discontinuous Galerkin method, where polynomial solutions are represented using a Taylor basis. A second order implicit method is applied for temporal discretization to the resulting ordinary differential equations. The resulting non-linear system of equations is solved at each time step using a pseudo-time marching approach. A newly developed fast, p -multigrid is then used to obtain the steady state solution to the pseudo-time system. The developed method is applied to compute a variety of unsteady subsonic viscous flow problems. The numerical results obtained indicate that the use of this implicit method leads to significant improvements in performance over its explicit counterpart, while without significant increase in memory requirements.
TL;DR: In this paper, the authors propose a method of adaptive time stepping, which adjusts the particle sub-time step to the need of the individual particle, to optimize the functionality of the EULER scheme.
Abstract: Numerical implementation schemes of drag force effects on Lagrangian particles can lead to instabilities or inefficiencies if static particle time stepping is used. Despite well known disadvantages, the programming structure of the underlying, C++ based, Lagrangian particle solver led to the choice of an explicit EULER, temporal discretization scheme. To optimize the functionality of the EULER scheme, this paper proposes a method of adaptive time stepping, which adjusts the particle sub time step to the need of the individual particle. A user definable adjustment between numerical stability and calculation efficiency is sought and a simple time stepping rule is presented. Furthermore a method to quantify numerical instability is devised and the importance of the characteristic particle relaxation time as numerical parameter is underlined. All derivations are being conducted for (non-)spherical particles and finally for a generalized drag force implementation. Important differences in spherical and non-sph...
Patent•
02 Jun 2009TL;DR: In this article, a computer implemented method for simulating a final pattern of a droplet of a fluid having a plurality of fluid properties is disclosed, which includes using lubrication equations to represent solute flow, diffusion and evaporation of the droplet on a substrate.
Abstract: A computer implemented method for simulating a final pattern of a droplet of a fluid having a plurality of fluid properties is disclosed. The method includes using lubrication equations to represent solute flow, diffusion and evaporation of a droplet on a substrate. The method further includes solving the lubrication equations through temporal discretization and spatial discretization; and deriving the final pattern of the droplet from results of the solving. The final pattern is stored on a computer readable medium.