scholarly journals Stability Analysis of Decoupled Time-stepping Schemes for the Specialized Conduction System/myocardium Coupled Problem in Cardiology

2017 ◽  
Vol 12 (5) ◽  
pp. 208-239 ◽  
Author(s):  
S. Mani Aouadi ◽  
W. Mbarki ◽  
N. Zemzemi
Keyword(s):  
Time Discretization ◽  
Conduction System ◽  
Splitting Schemes ◽  
Time Step ◽  
Time Stepping ◽  
Time Splitting ◽  
Coupling Terms ◽  
The Stability ◽  
Fully Coupled ◽  
Purkinje Network

The Purkinje network is the rapid conduction system in the heart. It ensures the physiological spread of the electrical wave in the ventricles. In this work, we consider a problem that models the coupling between the Purkinje network and the myocardium. We first prove the stability of the space semi-discretized problem. Then we present four different strategies for solving the Purkinje/ myocardium coupling. The strategies are based on different time discretization of the coupling terms. The first scheme is fully coupled, where the coupling terms are considered implicit. The second and the third schemes are based on Gauss-Seidel time-splitting schemes where one coupling term is considered explicit and the other is implicit. The last is a Jacobi-like time-splitting scheme where both coupling terms are considered explicit. Our main result is the proof of the stability of the three considered schemes under the same restriction on the time step. Moreover, we show that the energy of the problem is slightly affected by the time-splitting schemes. We illustrate the theoretical result by different numerical simulations in 2D. We also conduct 3D simulations using physiologically detailed ionic models.

Fast Conjugate Heat Transfer Simulation of Long Transient Flexible Operations Using Adaptive Time Stepping

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
R. Maffulli ◽  
L. He ◽  
P. Stein ◽  
G. Marinescu
Keyword(s):  
Heat Transfer ◽  
Conjugate Heat Transfer ◽  
Time Derivative ◽  
Computational Cost ◽  
Strong Impact ◽  
Time Step ◽  
Time Stepping ◽  
Adaptive Time ◽  
Fully Coupled ◽  
Adaptive Time Stepping

The emerging renewable energy market calls for more advanced prediction tools for turbine transient operations in fast startup/shutdown cycles. Reliable numerical analysis of such transient cycles is complicated by the disparity in time scales of the thermal responses in fluid and solid domains. Obtaining fully coupled time-accurate unsteady conjugate heat transfer (CHT) results under these conditions would require to march in both domains using the time-step dictated by the fluid domain: typically, several orders of magnitude smaller than the one required by the solid. This requirement has strong impact on the computational cost of the simulation as well as being potentially detrimental to the accuracy of the solution due to accumulation of round-off errors in the solid. A novel loosely coupled CHT methodology has been recently proposed, and successfully applied to both natural and forced convection cases that remove these requirements through a source-term based modeling (STM) approach of the physical time derivative terms in the relevant equations. The method has been shown to be numerically stable for very large time steps with adequate accuracy. The present effort is aimed at further exploiting the potential of the methodology through a new adaptive time stepping approach. The proposed method allows for automatic time-step adjustment based on estimating the magnitude of the truncation error of the time discretization. The developed automatic time stepping strategy is applied to natural convection cases under long (2000 s) transients: relevant to the prediction of turbine thermal loads during fast startups/shutdowns. The results of the method are compared with fully coupled unsteady simulations showing comparable accuracy with a significant reduction of the computational costs.

scholarly journals Spectral Direction Splitting Schemes for the Incompressible Navier-Stokes Equations

2011 ◽  
Vol 1 (3) ◽  
pp. 215-234 ◽  
Author(s):  
Lizhen Chen ◽  
Jie Shen ◽  
Chuanju Xu
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Splitting Schemes ◽  
Time Step ◽  
Order Of Accuracy ◽  
Navier Stokes Equations ◽  
Pressure Stabilization ◽  
Legendre Spectral Method ◽  
Direction Splitting ◽  
The Stability

AbstractWe propose and analyze spectral direction splitting schemes for the incompressible Navier-Stokes equations. The schemes combine a Legendre-spectral method for the spatial discretization and a pressure-stabilization/direction splitting scheme for the temporal discretization, leading to a sequence of one-dimensional elliptic equations at each time step while preserving the same order of accuracy as the usual pressure-stabilization schemes. We prove that these schemes are unconditionally stable, and present numerical results which demonstrate the stability, accuracy, and efficiency of the proposed methods.

scholarly journals Assessment of Hydrometeor Collection Rates from Exact and Approximate Equations. Part II: Numerical Bounding

2007 ◽  
Vol 46 (1) ◽  
pp. 82-96
Author(s):  
Brian J. Gaudet ◽  
Jerome M. Schmidt
Keyword(s):  
Mass Transfer ◽  
Exact Solution ◽  
Splitting Method ◽  
Time Discretization ◽  
Large Mass ◽  
Time Step ◽  
Transfer Rates ◽  
Mass Moment ◽  
Time Splitting ◽  
Approximate Equations

Abstract Past microphysical investigations, including Part I of this study, have noted that the collection equation, when applied to the interaction between different hydrometeor species, can predict large mass transfer rates, even when an exact solution is used. The fractional depletion in a time step can even exceed unity for the collected species with plausible microphysical conditions and time steps, requiring “normalization” by a microphysical scheme. Although some of this problem can be alleviated through the use of more moment predictions and hydrometeor categories, the question as to why such “overdepletion” can be predicted in the first place remains insufficiently addressed. It is shown through both physical and conceptual arguments that the explicit time discretization of the bulk collection equation for any moment is not consistent with a quasi-stochastic view of collection. The result, under certain reasonable conditions, is a systematic overprediction of collection, which can become a serious error in the prediction of higher-order moments in a bulk scheme. The term numerical bounding is used to refer to the use of a quasi-stochastically consistent formula that prevents fractional collections exceeding unity for any moments. Through examples and analysis it is found that numerical bounding is typically important in mass moment prediction for time steps exceeding approximately 10 s. The Poisson-based numerical bounding scheme is shown to be simple in application and conceptualization; within a straightforward idealization it completely corrects overdepletion while even being immune to the rediagnosis error of the time-splitting method. The scheme’s range of applicability and utility are discussed.

NUMERICAL GRADIENT FLOW DISCRETIZATION OF VISCOUS THIN FILMS ON CURVED GEOMETRIES

2013 ◽  
Vol 23 (05) ◽  
pp. 917-947 ◽  
Author(s):  
MARTIN RUMPF ◽  
ORESTIS VANTZOS
Keyword(s):  
Thin Films ◽  
Transport Equation ◽  
Degrees Of Freedom ◽  
Gradient Flow ◽  
Time Discretization ◽  
Time Step ◽  
Time Stepping ◽  
Exterior Calculus ◽  
Natural Time ◽  
Space Discretization

The evolution of a viscous thin film on a curved geometry is numerically approximated based on the natural time discretization of the underlying gradient flow. This discretization leads to a variational problem to be solved at each time step, which reflects the balance between the decay of the free (gravitational and surface) energy and the viscous dissipation. Both dissipation and energy are derived from a lubrication approximation for a small ratio between the characteristic film height and the characteristic length scale of the surface. The dissipation is formulated in terms of a corresponding flux field, whereas the energy primarily depends on the fluid volume per unit surface, which is a conserved quantity. These two degrees of freedom are coupled by the underlying transport equation. Hence, one is naturally led to a PDE-constrained optimization problem, where the variational time stepping problem has to be solved under the constraint described by the transport equation. For the space discretization a discrete exterior calculus approach is investigated. Various applications demonstrate the qualitative and quantitative behavior of one- and two-dimensional thin films on curved geometries.

scholarly journals Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3000
Author(s):  
Eric T. Chung ◽  
Yalchin Efendiev ◽  
Wing Tat Leung ◽  
Wenyuan Li
Keyword(s):  
Linear Equations ◽  
Nonlinear Problems ◽  
Time Discretization ◽  
Numerical Results ◽  
Explicit Methods ◽  
Implicit Methods ◽  
Time Step ◽  
Multiscale Problems ◽  
Time Stepping ◽  
Space Decomposition

This work continues a line of work on developing partially explicit methods for multiscale problems. In our previous works, we considered linear multiscale problems where the spatial heterogeneities are at the subgrid level and are not resolved. In these works, we have introduced contrast-independent, partially explicit time discretizations for linear equations. The contrast-independent, partially explicit time discretization divides the spatial space into two components: contrast dependent (fast) and contrast independent (slow) spaces defined via multiscale space decomposition. Following this decomposition, temporal splitting was proposed, which treats fast components implicitly and slow components explicitly. The space decomposition and temporal splitting are chosen such that they guarantees stability, and we formulated a condition for the time stepping. This condition was formulated as a condition on slow spaces. In this paper, we extend this approach to nonlinear problems. We propose a splitting approach and derive a condition that guarantees stability. This condition requires some type of contrast-independent spaces for slow components of the solution. We present numerical results and show that the proposed methods provide results similar to implicit methods with a time step that is independent of the contrast.

scholarly journals Braginskii viscosity on an unstructured, moving mesh accelerated with super-time-stepping

2019 ◽  
Vol 491 (2) ◽  
pp. 2919-2938 ◽  
Author(s):  
Thomas Berlok ◽  
Rüdiger Pakmor ◽  
Christoph Pfrommer
Keyword(s):  
Large Scale ◽  
Moving Mesh ◽  
Time Step ◽  
Time Stepping ◽  
Linearly Polarized ◽  
Anisotropic Transport ◽  
Cluster Mergers ◽  
The Stability ◽  
Large Scale Simulations ◽  
The Magnetic Field

ABSTRACT We present a method for efficiently modelling Braginskii viscosity on an unstructured, moving mesh. Braginskii viscosity, i.e. anisotropic transport of momentum with respect to the direction of the magnetic field, is thought to be of prime importance for studies of the weakly collisional plasma that comprises the intracluster medium (ICM) of galaxy clusters. Here, anisotropic transport of heat and momentum has been shown to have profound consequences for the stability properties of the ICM. Our new method for modelling Braginskii viscosity has been implemented in the moving mesh code arepo. We present a number of examples that serve to test the implementation and illustrate the modified dynamics found when including Braginskii viscosity in simulations. These include (but are not limited to) damping of fast magnetosonic waves, interruption of linearly polarized Alfvén waves by the firehose instability, and the inhibition of the Kelvin–Helmholtz instability by Braginskii viscosity. An explicit update of Braginskii viscosity is associated with a severe time-step constraint that scales with (Δx)2, where Δx is the grid size. In our implementation, this restrictive time-step constraint is alleviated by employing second-order accurate Runge–Kutta–Legendre super-time-stepping. We envision including Braginskii viscosity in future large-scale simulations of Kelvin–Helmholtz unstable cold fronts in cluster mergers and AGN-generated bubbles in central cluster regions.

scholarly journals Error Estimates of a Continuous Galerkin Time Stepping Method for Subdiffusion Problem

2021 ◽  
Vol 88 (3) ◽  
Author(s):  
Yuyuan Yan ◽  
Bernard A. Egwu ◽  
Zongqi Liang ◽  
Yubin Yan
Keyword(s):  
Piecewise Linear ◽  
Time Discretization ◽  
Convergence Order ◽  
Step Size ◽  
Time Step ◽  
Transform Method ◽  
Nonsmooth Data ◽  
Time Step Size ◽  
Time Stepping ◽  
Continuous Galerkin

AbstractA continuous Galerkin time stepping method is introduced and analyzed for subdiffusion problem in an abstract setting. The approximate solution will be sought as a continuous piecewise linear function in time t and the test space is based on the discontinuous piecewise constant functions. We prove that the proposed time stepping method has the convergence order $$O(\tau ^{1+ \alpha }), \, \alpha \in (0, 1)$$ O ( τ 1 + α ) , α ∈ ( 0 , 1 ) for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where $$\tau $$ τ is the time step size. This convergence order is higher than the convergence orders of the popular convolution quadrature methods (e.g., Lubich’s convolution methods) and L-type methods (e.g., L1 method), which have only $$O(\tau )$$ O ( τ ) convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity.

scholarly journals Super-Time-Stepping Scheme for Option Pricing

2020 ◽  
Vol 13 (13) ◽  
pp. 51-54
Author(s):  
Kedar Nath Uprety ◽  
Harithar Khanal ◽  
Ananta Upreti
Keyword(s):  
Option Pricing ◽  
Euler Method ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Stepping ◽  
Black Scholes ◽  
Explicit Finite Difference ◽  
The Stability ◽  
Explicit Finite Difference Method

We solve the Black - Scholes equation for option pricing numerically using an Explicit finite difference method. To overcome the stability restriction of the explicit scheme for parabolic partial differential equations in the time step size Courant-Friedrichs-Lewy (CFL) condition, we employ a Super Time Stepping (STS) strategy based on modified Chebyshev polynomial. The numerical results show that the STS scheme boasts of large efficiency gains compared to the standard explicit Euler method.

scholarly journals Artificial Compressibility Methods for the Incompressible Navier–Stokes Equations Using Lowest-Order Face-Based Schemes on Polytopal Meshes

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Riccardo Milani ◽  
Jérôme Bonelle ◽  
Alexandre Ern
Keyword(s):  
Stokes Equations ◽  
Time Discretization ◽  
Second Order ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Artificial Compressibility ◽  
Time Stepping ◽  
Monolithic Approach ◽  
Polytopal Meshes

Abstract We investigate artificial compressibility (AC) techniques for the time discretization of the incompressible Navier–Stokes equations. The space discretization is based on a lowest-order face-based scheme supporting polytopal meshes, namely discrete velocities are attached to the mesh faces and cells, whereas discrete pressures are attached to the mesh cells. This face-based scheme can be embedded into the framework of hybrid mixed mimetic schemes and gradient schemes, and has close links to the lowest-order version of hybrid high-order methods devised for the steady incompressible Navier–Stokes equations. The AC time-stepping uncouples at each time step the velocity update from the pressure update. The performances of this approach are compared against those of the more traditional monolithic approach which maintains the velocity-pressure coupling at each time step. We consider both first-order and second-order time schemes and either an implicit or an explicit treatment of the nonlinear convection term. We investigate numerically the CFL stability restriction resulting from an explicit treatment, both on Cartesian and polytopal meshes. Finally, numerical tests on large 3D polytopal meshes highlight the efficiency of the AC approach and the benefits of using second-order schemes whenever accurate discrete solutions are to be attained.

scholarly journals A Time-Stepping DPG Scheme for the Heat Equation

2017 ◽  
Vol 17 (2) ◽  
pp. 237-252 ◽  
Author(s):  
Thomas Führer ◽  
Norbert Heuer ◽  
Jhuma Sen Gupta
Keyword(s):  
Heat Equation ◽  
Order Approximation ◽  
Test Functions ◽  
Optimal Test ◽  
Time Step ◽  
Time Stepping ◽  
Backward Euler ◽  
The Stability ◽  
Convergence Orders ◽  
Theoretical Results

AbstractWe introduce and analyze a discontinuous Petrov–Galerkin method with optimal test functions for the heat equation. The scheme is based on the backward Euler time stepping and uses an ultra-weak variational formulation at each time step. We prove the stability of the method for the field variables (the original unknown and its gradient weighted by the square root of the time step) and derive a Céa-type error estimate. For low-order approximation spaces this implies certain convergence orders when time steps are not too small in comparison with mesh sizes. Some numerical experiments are reported to support our theoretical results.

Sign in / Sign up

Export Citation Format

Share Document