Implicit-Explicit Runge-Kutta Schemes for the Boltzmann-Poisson System for Semiconductors

AbstractIn this paper we develop a class of Implicit-Explicit Runge-Kutta schemes for solving the multi-scale semiconductor Boltzmann equation. The relevant scale which characterizes this kind of problems is the diffusive scaling. This means that, in the limit of zero mean free path, the system is governed by a drift-diffusion equation. Our aim is to develop a method which accurately works for the different regimes encountered in general semiconductor simulations: the kinetic, the intermediate and the diffusive one. Moreover, we want to overcome the restrictive time step conditions of standard time integration techniques when applied to the solution of this kind of phenomena without any deterioration in the accuracy. As a result, we obtain high order time and space discretization schemes which do not suffer from the usual parabolic stiffness in the diffusive limit. We show different numerical results which permit to appreciate the performances of the proposed schemes.

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Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

Open Mathematics ◽

10.2478/s11533-013-0252-6 ◽

2013 ◽

Vol 11 (8) ◽

Cited By ~ 3

Author(s):

Mehmet Ersoy ◽

Frédéric Golay ◽

Lyudmyla Yushchenko

Keyword(s):

Shock Tube ◽

Conservation Laws ◽

Entropy Production ◽

Hyperbolic Conservation Laws ◽

Time Integration ◽

Posteriori Error ◽

Entropy Inequality ◽

Numerical Density ◽

Time Step ◽

High Order Time

AbstractWe propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). This density is used as an a posteriori error which provides information if the mesh should be refined in the regions where discontinuities occur or coarsened in the regions where the solution remains smooth. As due to the Courant-Friedrich-Levy stability condition the time step is restricted and leads to time consuming simulations, we propose a local time stepping algorithm. We also use high order time extensions applying the Adams-Bashforth time integration technique as well as the second order linear reconstruction in space. We numerically investigate the efficiency of the scheme through several test cases: Sod’s shock tube problem, Lax’s shock tube problem and the Shu-Osher test problem.

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Conservative Transport Schemes for Spherical Geodesic Grids: High-Order Flux Operators for ODE-Based Time Integration

Monthly Weather Review ◽

10.1175/mwr-d-10-05056.1 ◽

2011 ◽

Vol 139 (9) ◽

pp. 2962-2975 ◽

Cited By ~ 55

Author(s):

William C. Skamarock ◽

Almut Gassmann

Keyword(s):

Fourth Order ◽

Time Integration ◽

Higher Order ◽

Second Order ◽

Time Step ◽

Diffusion Term ◽

The Third ◽

Integration Schemes ◽

Order Scheme ◽

Runge Kutta

Higher-order finite-volume flux operators for transport algorithms used within Runge–Kutta time integration schemes on irregular Voronoi (hexagonal) meshes are proposed and tested. These operators are generalizations of third- and fourth-order operators currently used in atmospheric models employing regular, orthogonal rectangular meshes. Two-dimensional least squares fit polynomials are used to evaluate the higher-order spatial derivatives needed to cancel the leading-order truncation error terms of the standard second-order centered formulation. Positive definite or monotonic behavior is achieved by applying an appropriate limiter during the final Runge–Kutta stage within a given time step. The third- and fourth-order formulations are evaluated using standard transport tests on the sphere. The new schemes are more accurate and significantly more efficient than the standard second-order scheme and other schemes in the literature examined by the authors. The third-order formulation is equivalent to the fourth-order formulation plus an additional diffusion term that is proportional to the Courant number. An optimal value for the coefficient scaling this diffusion term is chosen based on qualitative evaluation of the test results. Improvements using the higher-order scheme in place of the traditional second-order centered approach are illustrated within 3D unstable baroclinic wave simulations produced using two global nonhydrostatic models employing spherical Voronoi meshes.

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Research in Numerical Simulation of Lager Multi-Scale Water Conveyance Tunnel Based on Mixed Time Integration Method

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.203.432 ◽

2012 ◽

Vol 203 ◽

pp. 432-437

Author(s):

Jun Jie Zhao ◽

Yan Zhi Yang

Keyword(s):

Numerical Simulation ◽

Computational Efficiency ◽

Integration Method ◽

Time Integration ◽

Scale Model ◽

Integration Time ◽

Time Step ◽

Detailed Model ◽

Multi Scale ◽

Time Integration Method

The global integration time step of multi-scale model is subject to local detailed model, resulting in lower computational efficiency. Mixed time integration method uses different integration time-step in different scale model, it can effectively avoid the above problem. Rest on a large-scale water tunnel under construction, in order to achieve the synchronization of the global and the local simulation, the paper establishes multi-scale finite element model of the tunnel, and calculate it by mixed time integration method. The final calculation and analysis show that the algorithm can guarantee the computational accuracy of multi-scale numerical simulation, and can effectively improve the computational efficiency, it can also provide references for related tunnel project.

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Implicit–explicit (IMEX) Runge–Kutta methods for non-hydrostatic atmospheric models

Geoscientific Model Development ◽

10.5194/gmd-11-1497-2018 ◽

2018 ◽

Vol 11 (4) ◽

pp. 1497-1515 ◽

Cited By ~ 11

Author(s):

David J. Gardner ◽

Jorge E. Guerra ◽

François P. Hamon ◽

Daniel R. Reynolds ◽

Paul A. Ullrich ◽

...

Keyword(s):

Acoustic Waves ◽

Time Integration ◽

Coupled System ◽

Atmospheric Model ◽

Step Size ◽

Time Step ◽

Solution Quality ◽

Time Step Size ◽

Runge Kutta ◽

The Impact

Abstract. The efficient simulation of non-hydrostatic atmospheric dynamics requires time integration methods capable of overcoming the explicit stability constraints on time step size arising from acoustic waves. In this work, we investigate various implicit–explicit (IMEX) additive Runge–Kutta (ARK) methods for evolving acoustic waves implicitly to enable larger time step sizes in a global non-hydrostatic atmospheric model. The IMEX formulations considered include horizontally explicit – vertically implicit (HEVI) approaches as well as splittings that treat some horizontal dynamics implicitly. In each case, the impact of solving nonlinear systems in each implicit ARK stage in a linearly implicit fashion is also explored.The accuracy and efficiency of the IMEX splittings, ARK methods, and solver options are evaluated on a gravity wave and baroclinic wave test case. HEVI splittings that treat some vertical dynamics explicitly do not show a benefit in solution quality or run time over the most implicit HEVI formulation. While splittings that implicitly evolve some horizontal dynamics increase the maximum stable step size of a method, the gains are insufficient to overcome the additional cost of solving a globally coupled system. Solving implicit stage systems in a linearly implicit manner limits the solver cost but this is offset by a reduction in step size to achieve the desired accuracy for some methods. Overall, the third-order ARS343 and ARK324 methods performed the best, followed by the second-order ARS232 and ARK232 methods.

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A Two-Stage Fourth-Order Multimoment Global Shallow-Water Model on the Cubed Sphere

Monthly Weather Review ◽

10.1175/mwr-d-20-0004.1 ◽

2020 ◽

Vol 148 (10) ◽

pp. 4267-4279

Author(s):

Yuzhang Che ◽

Chungang Chen ◽

Feng Xiao ◽

Xingliang Li ◽

Xueshun Shen

Keyword(s):

Shallow Water ◽

Fourth Order ◽

Time Integration ◽

Water Model ◽

Shallow Water Model ◽

Time Step ◽

Two Stage ◽

Order Scheme ◽

Runge Kutta ◽

Cubed Sphere

AbstractA new multimoment global shallow-water model on the cubed sphere is proposed by adopting a two-stage fourth-order Runge–Kutta time integration. Through calculating the values of predicted variables at half time step t = tn + (1/2)Δt by a second-order formulation, a fourth-order scheme can be derived using only two stages within one time step. This time integration method is implemented in our multimoment global shallow-water model to build and validate a new and more efficient numerical integration framework for dynamical cores. As the key task, the numerical formulation for evaluating the derivatives in time has been developed through the Cauchy–Kowalewski procedure and the spatial discretization of the multimoment finite-volume method, which ensures fourth-order accuracy in both time and space. Several major benchmark tests are used to verify the proposed numerical framework in comparison with the existing four-stage fourth-order Runge–Kutta method, which is based on the method of lines framework. The two-stage fourth-order scheme saves about 30% of the computational cost in comparison with the four-stage Runge–Kutta scheme for global advection and shallow-water models. The proposed two-stage fourth-order framework offers a new option to develop high-performance time marching strategy of practical significance in dynamical cores for atmospheric and oceanic models.

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Implicit-explicit (IMEX) Runge-Kutta methods for non-hydrostatic atmospheric models

10.5194/gmd-2017-285 ◽

2017 ◽

Cited By ~ 1

Author(s):

David J. Gardner ◽

Jorge E. Guerra ◽

François P. Hamon ◽

Daniel R. Reynolds ◽

Paul A. Ullrich ◽

...

Keyword(s):

Acoustic Waves ◽

Time Integration ◽

Coupled System ◽

Atmospheric Model ◽

Step Size ◽

Time Step ◽

Solution Quality ◽

Time Step Size ◽

Runge Kutta ◽

The Impact

Abstract. The efficient simulation of non-hydrostatic atmospheric dynamics requires time integration methods capable of overcoming the explicit stability constraints on time step size arising from acoustic waves. In this work we investigate various implicit-explicit (IMEX) additive Runge-Kutta (ARK) methods for evolving acoustic waves implicitly to enable larger time step sizes in a global non-hydrostatic atmospheric model. The IMEX formulations considered include horizontally implicit-vertically implicit (HEVI) approaches as well as splittings that treat some horizontal dynamics implicitly. In each case the impact of solving nonlinear systems in each implicit ARK stage in a linearly implicit fashion is also explored. The accuracy and efficiency of the IMEX splittings, ARK methods, and solver options are evaluated on a gravity wave and baroclinic wave test case. HEVI splittings that treat some vertical dynamics explicitly do not show a benefit in solution quality or run time over the most implicit HEVI formulation. While splittings that implicitly evolve some horizontal dynamics increase the maximum stable step size of a method, the gains are insufficient to overcome the additional cost of solving a globally coupled system. Solving implicit stage systems in a linearly implicit manner limits the solver cost but this is offset by a reduction in step size to achieve the desired accuracy. Overall, the third order ARS343 and ARK324 methods performed the best, followed by the second order ARS232 and ARK232 methods.

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OPTIMIZED ARTIFICIAL DISSIPATION TERMS FOR ENHANCED STABILITY LIMITS

Journal of Computational Acoustics ◽

10.1142/s0218396x07003329 ◽

2007 ◽

Vol 15 (02) ◽

pp. 235-253

Author(s):

HERWIG A. GROGGER

Keyword(s):

Finite Difference ◽

Time Integration ◽

Spatial Discretization ◽

Time Step ◽

Artificial Dissipation ◽

Finite Difference Approximations ◽

Difference Approximations ◽

Runge Kutta ◽

Stability Limits ◽

Enhanced Stability

Finite difference approximations for the convection equation are developed, which exhibit enhanced stability limits for explicit Runge–Kutta integration. Stability limits are increased by adding artificial dissipation terms, which are optimized to yield greatest stable time steps. For the artificial dissipation terms, symmetric finite difference approximations of even-order derivatives are used with differencing stencils equal to the convective stencils. The spatial discretization inclusive of the added dissipation term is shown to be consistent with a first derivative. The formal order of accuracy in space is decreased by one order, while the order of time integration is not affected. As a result, the time step limits of originally stable Runge–Kutta integration is increased, for some combinations of spatial discretization and time integration by a factor of two. Algorithms, which are unstable without damping are stabilized. The dispersion properties of the algorithms are not influenced by the proposed damping terms. Spectral analysis of the algorithms show very low dissipation error for dimensionless wave numbers k Δ x < 0.5. Stability conditions based on von Neumann stability analysis are given for the proposed schemes for explicit Runge–Kutta time integration of orders up to ten.

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A Fully Implicit Finite Volume Scheme for a Seawater Intrusion Problem in Coastal Aquifers

Water ◽

10.3390/w12061639 ◽

2020 ◽

Vol 12 (6) ◽

pp. 1639

Author(s):

Abdelkrim Aharmouch ◽

Brahim Amaziane ◽

Mustapha El Ossmani ◽

Khadija Talali

Keyword(s):

Finite Volume ◽

Coastal Aquifers ◽

Nonlinear Partial Differential Equations ◽

Time Integration ◽

Mass Conservation ◽

Coupled System ◽

Finite Volume Scheme ◽

Time Step ◽

Volume Scheme ◽

Fully Implicit

We present a numerical framework for efficiently simulating seawater flow in coastal aquifers using a finite volume method. The mathematical model consists of coupled and nonlinear partial differential equations. Difficulties arise from the nonlinear structure of the system and the complexity of natural fields, which results in complex aquifer geometries and heterogeneity in the hydraulic parameters. When numerically solving such a model, due to the mentioned feature, attempts to explicitly perform the time integration result in an excessively restricted stability condition on time step. An implicit method, which calculates the flow dynamics at each time step, is needed to overcome the stability problem of the time integration and mass conservation. A fully implicit finite volume scheme is developed to discretize the coupled system that allows the use of much longer time steps than explicit schemes. We have developed and implemented this scheme in a new module in the context of the open source platform DuMu X . The accuracy and effectiveness of this new module are demonstrated through numerical investigation for simulating the displacement of the sharp interface between saltwater and freshwater in groundwater flow. Lastly, numerical results of a realistic test case are presented to prove the efficiency and the performance of the method.

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A Middleware-Based Approach for Multi-Scale Mobility Simulation

Future Internet ◽

10.3390/fi13020022 ◽

2021 ◽

Vol 13 (2) ◽

pp. 22

Author(s):

Xavier Boulet ◽

Mahdi Zargayouna ◽

Gérard Scemama ◽

Fabien Leurent

Keyword(s):

Modeling And Simulation ◽

Transportation Networks ◽

External Control ◽

Independent Mobility ◽

Time Step ◽

Time Representation ◽

Multi Scale ◽

Networks Analysis ◽

Maximal Benefit ◽

Do So

Modeling and simulation play an important role in transportation networks analysis. In the literature, authors have proposed many traffic and mobility simulations, with different features and corresponding to different contexts and objectives. They notably consider different scales of simulations. The scales refer to the represented entities, as well as to the space and the time representation of the transportation environment. However, we often need to represent different scales in the same simulation, for instance to represent a neighborhood interacting with a wider region. In this paper, we advocate for the reuse of existing simulations to build a new multi-scale simulation. To do so, we propose a middleware model to couple independent mobility simulations, working at different scales. We consider all the necessary processing and workflow to allow for a coherent orchestration of these simulations. We also propose a prototype implementation of the middleware. The results show that such a middleware is capable of creating a new multi-scale mobility simulation from existing ones, while minimizing the incoherence between them. They also suggest that, to have a maximal benefit from the middleware, existing mobility simulation platforms should allow for an external control of the simulations, allowing for executing a time step several times if necessary.

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Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽

10.3390/math9131483 ◽

2021 ◽

Vol 9 (13) ◽

pp. 1483

Author(s):

Shanqin Chen

Keyword(s):

Large Time ◽

Hyperbolic Equations ◽

Strong Stability ◽

Matrix Exponential ◽

Linear Component ◽

Nonlinear Hyperbolic Equations ◽

Time Step ◽

Integrating Factor ◽

The Matrix ◽

Runge Kutta

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

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