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.

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 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 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.

An Efficient Computing Explicit Method for Structural Dynamics

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Shuenn-Yih Chang
Keyword(s):  
Structural Dynamics ◽  
Unconditional Stability ◽  
Explicit Methods ◽  
Explicit Method ◽  
Implicit Methods ◽  
Time Step ◽  
Integration Algorithm ◽  
Structural Dynamic ◽  
Linear Elastic ◽  
Linearized Stability

An integration algorithm, which integrates the most important advantage of explicit methods of the explicitness of each time step and that of implicit methods of the possibility of unconditional stability, is presented herein. This algorithm is analytically shown to be unconditionally stable for any linear elastic and nonlinear systems except for the instantaneous stiffness hardening systems with the instantaneous degree of nonlinearity larger than 43 based on a linearized stability analysis. Hence, its stability property is better than the previously published algorithm (Chang, 2007, “Improved Explicit Method for Structural Dynamics,” J. Eng. Mech., 133(7), pp. 748–760), which is only conditionally stable for instantaneous stiffness hardening systems although it also possesses unconditional stability for linear elastic and any instantaneous stiffness softening systems. Due to the explicitness of each time step, the possibility of unconditional stability, and comparable accuracy, the proposed algorithm is very promising for a general structural dynamic problem, where only the low frequency responses are of interest since it consumes much less computational efforts when compared with explicit methods, such as the Newmark explicit method, and implicit methods, such as the constant average acceleration method.

Efficient Time-Stepping/Spectral Methods for the Navier-Stokes-Nernst-Planck-Poisson Equations

2017 ◽  
Vol 21 (5) ◽  
pp. 1408-1428 ◽  
Author(s):  
Xiaoling Liu ◽  
Chuanju Xu
Keyword(s):  
Stability Condition ◽  
Time Discretization ◽  
Equation System ◽  
Second Order ◽  
Navier Stokes ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
First Order ◽  
Time Stepping

AbstractThis paper is concerned with numerical methods for the Navier-Stokes-Nernst-Planck-Poisson equation system. The main goal is to construct and analyze some stable time stepping schemes for the time discretization and use a spectral method for the spatial discretization. The main contribution of the paper includes: 1) an useful stability inequality for the weak solution is derived; 2) a first order time stepping scheme is constructed, and the non-negativity of the concentration components of the discrete solution is proved. This is an important property since the exact solution shares the same property. Moreover, the stability of the scheme is established, together with a stability condition on the time step size; 3) a modified first order scheme is proposed in order to decouple the calculation of the velocity and pressure in the fluid field. This new scheme equally preserves the non-negativity of the discrete concentration solution, and is stable under a similar stability condition; 4) a stabilization technique is introduced to make the above mentioned schemes stable without restriction condition on the time step size; 5) finally we construct a second order finite difference scheme in time and spectral discretization in space. The numerical tests carried out in the paper show that all the proposed schemes possess some desirable properties, such as conditionally/unconditionally stability, first/second order convergence, non-negativity of the discrete concentrations, and so on.

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.

scholarly journals All Speed Scheme for the Low Mach Number Limit of the Isentropic Euler Equations

2011 ◽  
Vol 10 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Pierre Degond ◽  
Min Tang
Keyword(s):  
Mach Number ◽  
Euler Equations ◽  
Computational Cost ◽  
Time Discretization ◽  
Implicit Time ◽  
Elliptic Type ◽  
Implicit Methods ◽  
Time Step ◽  
Low Mach Number ◽  
Isentropic Euler Equations

AbstractAn all speed scheme for the Isentropic Euler equations is presented in this paper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficulty in the low Mach regime. The key idea of our all speed scheme is the special semi-implicit time discretization, in which the low Mach number stiff term is divided into two parts, one being treated explicitly and the other one implicitly. Moreover, the flux of the density equation is also treated implicitly and an elliptic type equation is derived to obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared with previous semi-implicit methods, firstly, nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs to be solved implicitly which reduces much computational cost. We develop this semi-implicit time discretization in the framework of a first order Local Lax-Friedrichs (or Rusanov) scheme and numerical tests are displayed to demonstrate its performances.

scholarly journals Semi-Lagrangian Subgrid Reconstruction for Advection-Dominant Multiscale Problems with Rough Data

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
Konrad Simon ◽  
Jörn Behrens
Keyword(s):  
Nonlinear Problems ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Multiscale Methods ◽  
Galerkin Methods ◽  
Standard Methods ◽  
Multiscale Problems ◽  
Eulerian Method ◽  
Lower Order Terms ◽  
New Framework

AbstractWe introduce a new framework of numerical multiscale methods for advection-dominated problems motivated by climate sciences. Current numerical multiscale methods (MsFEM) work well on stationary elliptic problems but have difficulties when the model involves dominant lower order terms. Our idea to overcome the associated difficulties is a semi-Lagrangian based reconstruction of subgrid variability into a multiscale basis by solving many local inverse problems. Globally the method looks like a Eulerian method with multiscale stabilized basis. We show example runs in one and two dimensions and a comparison to standard methods to support our ideas and discuss possible extensions to other types of Galerkin methods, higher dimensions and nonlinear problems.

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 Goal-Oriented Error Estimation for the Fractional Step Theta Scheme

2014 ◽  
Vol 14 (2) ◽  
pp. 203-230 ◽  
Author(s):  
Dominik Meidner ◽  
Thomas Richter
Keyword(s):  
Time Discretization ◽  
Posteriori Error ◽  
Numerical Dissipation ◽  
Error Estimator ◽  
Fractional Step ◽  
Weighted Residual Method ◽  
Time Stepping ◽  
Backward Euler ◽  
Galerkin Formulation ◽  
Theta Method

Abstract. In this work, we derive a goal-oriented a posteriori error estimator for the error due to time-discretization of nonlinear parabolic partial differential equations by the fractional step theta method. This time-stepping scheme is assembled by three steps of the general theta method, that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson method. Further, by combining three substeps of the theta time-stepping scheme, the fractional step theta time-stepping scheme is derived. It possesses highly desired stability and numerical dissipation properties and is second order accurate. The derived error estimator is based on a Petrov–Galerkin formulation that is up to a numerical quadrature error equivalent to the theta time-stepping scheme. The error estimator is assembled as one weighted residual term given by the dual weighted residual method and one additional residual estimating the Galerkin error between time-stepping scheme and Petrov–Galerkin formulation.

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