Numerical approximation of Turing patterns in electrodeposition by ADI methods

In this paper we study the numerical approximation of Turing patterns corresponding to steady state solutions of a PDE system of reaction–diffusion equations modeling an electrodeposition process. We apply the Method of Lines (MOL) and describe the semi-discretization by high order finite differences in space given by the Extended Central Difference Formulas (ECDFs) that approximate Neumann boundary conditions (BCs) with the same accuracy. We introduce a test equation to describe the interplay between the diffusion and the reaction time scales. We present a stability analysis of a selection of time-integrators (IMEX 2-SBDF method, Crank–Nicolson (CN), Alternating Direction Implicit (ADI) method) for the test equation as well as for the Schnakenberg model, prototype of nonlinear reaction–diffusion systems with Turing patterns. Eventually, we apply the ADI-ECDF schemes to solve the electrodeposition model until the stationary patterns (spots & worms and only spots) are reached. We validate the model by comparison with experiments on Cu film growth by electrodeposition.     

Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows

Magnetohydrodynamic (MHD) flows are governed by Navier–Stokes equations coupled with Maxwell equations through coupling terms. We prove the unconditional stability of a partitioned method for the evolutionary full MHD equations, at high magnetic Reynolds number, written in the Elsässer variables. The method we analyze is a first-order one-step scheme, which consists of implicit discretization of the subproblem terms and explicit discretization of the coupling terms.     

VARIABLE STEP-SIZE IMPLICIT-EXPLICIT LINEAR MULTISTEP METHODS FOR TIME-DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS

Implicit-explicit （IMEX） linear multistep methods are popular techniques for solving partial differential equations （PDEs） with terms of different types. While fixed timestep versions of such schemes have been developed and studied, implicit-explicit schemes also naturally arise in general situations where the temporal smoothness of the solution changes. In this paper we consider easily implementable variable step-size implicit-explicit （VSIMEX） linear multistep methods for time-dependent PDEs. Families of order-p, pstep VSIMEX schemes are constructed and analyzed, where p ranges from 1 to 4. The corresponding schemes are simple to implement and have the property that they reduce to the classical IMEX schemes whenever constant time step-sizes are imposed. The methods are validated on the Burgers＇ equation. These results demonstrate that by varying the time step-size, VSIMEX methods can outperform their fixed time step counterparts while still maintaining good numerical behavior.     

A hybrid time stepping scheme combining explicit Runge–Kutta with implicit LU‐SGS for Navier–Stokes equations

A hybrid time stepping scheme is developed and implemented by a combination of explicit Runge–Kutta with implicit LU‐SGS scheme at the level of system matrix. In this method, the explicit scheme is applied to those grid cells of blocks that have large local time steps; meanwhile, the implicit scheme is applied to other grid cells of blocks that have smaller allowable local time steps in the same flow field. As a result, the discretized governing equations can be expressed as a compound of explicit and implicit matrix operator. The proposed method has been used to compute the steady transonic turbulent flow over the RAE 2822 airfoil. The numerical results are found to be in excellent agreement with the experimental data. In the validation case, the present scheme saved at least 50% of the memory resources compared with the fully implicit LU‐SGS. Copyright © 2015 John Wiley & Sons, Ltd.     

Uncoupling evolutionary groundwater‐surface water flows using the Crank–Nicolson Leapfrog method

Consider an incompressible fluid in a region Ω_f flowing both ways across an interface into a porous media domain Ω_p saturated with the same fluid. The physical processes in each domain have been well studied and are described by the Stokes equations in the fluid region and the Darcy equations in the porous media region. Taking the interfacial conditions into account produces a system with an exactly skew symmetric coupling. Spatial discretization by finite element method and time discretization by Crank–Nicolson LeapFrog give a second‐order partitioned method requiring only one Stokes and one Darcy subphysics and subdomain solver per time step for the fully evolutionary Stokes‐Darcy problem. Analysis of this method leads to a time step condition sufficient for stability and convergence. Numerical tests verify predicted rates of convergence; however, stability tests reveal the problem of growth of numerical noise in unstable modes in some cases. In such instances, the addition of time filters adds stability. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A second order self-consistent IMEX method for radiation hydrodynamics

We present a second order self-consistent implicit/explicit (methods that use the combination of implicit and explicit discretizations are often referred to as IMEX (implicit/explicit) methods ,  and ) time integration technique for solving radiation hydrodynamics problems. The operators of the radiation hydrodynamics are splitted as such that the hydrodynamics equations are solved explicitly making use of the capability of well-understood explicit schemes. On the other hand, the radiation diffusion part is solved implicitly. The idea of the self-consistent IMEX method is to hybridize the implicit and explicit time discretizations in a nonlinearly consistent way to achieve second order time convergent calculations. In our self-consistent IMEX method, we solve the hydrodynamics equations inside the implicit block as part of the nonlinear function evaluation making use of the Jacobian-free Newton Krylov (JFNK) method ,  and . This is done to avoid order reductions in time convergence due to the operator splitting. We present results from several test calculations in order to validate the numerical order of our scheme. For each test, we have established second order time convergence.     

An adaptive implicit–explicit scheme for the DNS and LES of compressible flows on unstructured grids

An adaptive implicit–explicit scheme for Direct Numerical Simulation (DNS) and Large-Eddy Simulation (LES) of compressible turbulent flows on unstructured grids is developed. The method uses a node-based finite-volume discretization with Summation-by-Parts (SBP) property, which, in conjunction with Simultaneous Approximation Terms (SAT) for imposing boundary conditions, leads to a linearly stable semi-discrete scheme. The solution is marched in time using an Implicit–Explicit Runge–Kutta (IMEX-RK) time-advancement scheme. A novel adaptive algorithm for splitting the system into implicit and explicit sets is developed. The method is validated using several canonical laminar and turbulent flows. Load balance for the new scheme is achieved by a dual-constraint, domain decomposition algorithm. The scalability and computational efficiency of the method is investigated, and memory savings compared with a fully implicit method is demonstrated. A notable reduction of computational costs compared to both fully implicit and fully explicit schemes is observed.     

IMEX method convergence for a parabolic equation

Although implicit-explicit (IMEX) methods for approximating solutions to semilinear parabolic equations are relatively standard, most recent works examine the case of a fully discretized model. We show that by discretizing time only, one can obtain an elementary convergence result for an implicit-explicit method. This convergence result is strong enough to imply existence and uniqueness of solutions to a class of semilinear parabolic equations.     

Stability of implicit-explicit linear multistep methods for ordinary and delay differential equations

Stability properties of implicit-explicit (IMEX) linear multistep methods for ordinary and delay differential equations are analyzed on the basis of stability regions defined by using scalar test equations. The analysis is closely related to the stability analysis of the standard linear multistep methods for delay differential equations. A new second-order IMEX method which has approximately the same stability region as that of the IMEX Euler method, the simplest IMEX method of order 1, is proposed. Some numerical results are also presented which show superiority of the new method.      

Modelling and numerical methods for the dynamics of impurities in a gas

The dynamics of a mixture of impurities in a gas can be represented by a system of linear Boltzmann equations for hard spheres. We assume that the background is in thermodynamic equilibrium and that the polluting particles are sufficiently few (in comparison with the background molecules) to admit that there are no collisions among couples of them. In order to derive non‐trivial hydrodynamic models, we close the Euler system around local Maxwellian's which are not equilibrium states. The kinetic model is solved by using a Monte Carlo method, the hydrodynamic one by implicit–explicit Runge–Kutta schemes with weighted essentially non‐oscillatory reconstruction (J. Sci. Comput. 2005; 25 (1–2):129–155). Several numerical tests are then computed in order to compare the results obtained with the kinetic and the hydrodynamic models. Copyright © 2007 John Wiley & Sons, Ltd.