Explicit Time Discretization Research Articles

A Stable and High Accurate Numerical Simulator for Convection-Diffusion Equation

This study presents a simulator to obtain numerical solution of convection-diffusion equation. It includes explicit, fully implicit and semi-implicit time discretization techniques. Although the explicit method is efficient for some simple conduction problems, it needs stability criteria. Otherwise, it is unstable especially for high Courant number and large time step size. When using the explicit method, the time step size is chosen with care to get well-posed numerical solution. This requirement sets a serious constraint for the explicit method because choosing small time step size causes the simulation time to become quite long. Even though the implicit method is stable for large time step size and high Courant number, it leads to numerical dispersion like the explicit method. On the other hand, second-order accurate semi-implicit method significantly reduces numerical dispersion. In addition to time discretization techniques, this simulator contains several space discretization methods such as first-order upstream and UMIST (University of Manchester Institute of Science and Technology) techniques. The proposed numerical simulator is suitable for easily using the different combinations of time and space discretization methods. Secondly, this study is to propose the use of semi-implicit time discretization technique with UMIST space discretization method to minimize numerical dispersion and suppress unphysical oscillation. In spite of the fact that the UMIST method suppresses unphysical oscillation, it causes a small and undesired oscillation at flood front for very large Courant number. Third objective of this study is to propose minor modification on the UMIST method to eliminate this unphysical oscillation.

Meshless approach to the large-eddy simulation of the continuous casting process

The paper discusses the numerical solution of the large-eddy formulation for modelling the turbulent fluid flow with solidification in the continuous casting of steel. This industrially relevant problem is solved by a meshless method for the first time. The solid-liquid system is formulated within the continuum mixture formulation of the governing, mass momentum and energy equations. The Darcy porous media model describes the mushy region. The large-eddy formulation is based on the Smagorinsky model for sub-grid scale viscosity and the van Driest damping function. The synthetic turbulence is prescribed at the molten steel inlet. The generated turbulent fluctuations are isotopic and correlated in time through an asymmetric time filter. Explicit time discretisation, combined with meshless local radial basis function collocation method, is employed. Five-noded subdomains, non-uniform node arrangement and multiquadrics shape functions are used for solving a two-dimensional model. The sensitivity to different node arrangements, time steps, inlet temperature, and velocity boundary conditions is elaborated and successfully verified by comparison with our previously published meshless k − ε turbulence model of the same process.

A comparison of the stability and performance of depth-integrated ice-dynamics solvers

Abstract. In the last decade, the number of ice-sheet models has increased substantially, in line with the growth of the glaciological community. These models use solvers based on different approximations of ice dynamics. In particular, several depth-integrated dynamics solvers have emerged as fast solvers capable of resolving the relevant physics of ice sheets at the continental scale. However, the numerical stability of these schemes has not been studied systematically to evaluate their effectiveness in practice. Here we focus on three such solvers, the so-called Hybrid, L1L2-SIA and DIVA solvers, as well as the well-known SIA and SSA solvers as boundary cases. We investigate the numerical stability of these solvers as a function of grid resolution and the state of the ice sheet for an explicit time discretization scheme of the mass conservation step. Under simplified conditions with constant viscosity, the maximum stable time step of the Hybrid solver, like the SIA solver, has a quadratic dependence on grid resolution. In contrast, the DIVA solver has a maximum time step that is independent of resolution as the grid becomes increasingly refined, like the SSA solver. A simple 1D implementation of the L1L2-SIA solver indicates that it should behave similarly, but in practice, the complexity of its implementation appears to restrict its stability. In realistic simulations of the Greenland Ice Sheet with a nonlinear rheology, the DIVA and SSA solvers maintain superior numerical stability, while the SIA, Hybrid and L1L2-SIA solvers show markedly poorer performance. At a grid resolution of Δx=4 km, the DIVA solver runs approximately 20 times faster than the Hybrid and L1L2-SIA solvers as a result of a larger stable time step. Our analysis shows that as resolution increases, the ice-dynamics solver can act as a bottleneck to model performance. The DIVA solver emerges as a clear outlier in terms of both model performance and its representation of the ice-flow physics itself.

Contrast-Independent Partially Explicit Time Discretizations for Quasi Gas Dynamics

In the paper, we study a design and stability of contrast-independent partially explicit time discretizations for Quasi-Gas-Dynamics (QGD) Equations in multiscale high-contrast media. In our previous works, we have introduced contrast-independent partially explicit time discretizations. In this paper, we extend these ideas to multiscale QGD problems. Because of high contrast, explicit methods require a very small time stepping. By designing appropriate spatial splitting and temporal splitting, partially explicit methods remove this constraint. The proposed partially explicit time discretization consists of two steps. First, we split the space into contrast dependent (fast) and contrast independent (slow) components on a coarse grid that is much larger compared to spatial heterogeneities. Secondly, we design a temporal splitting algorithm in a such way that it is stable and the time step is independent of the contrast and only depends on the coarse mesh size. Using proposed method, a few degrees of freedom are treated implicitly and the approach is mostly explicit. We prove that the proposed splitting is unconditionally stable under some suitable conditions formulated for the second space (slow). We present numerical results and show that the proposed methods provide results similar to implicit methods with the time step that is independent of the contrast.

High order semi-implicit weighted compact nonlinear scheme for the full compressible Euler system at all Mach numbers

A high order semi-implicit weighted compact nonlinear scheme (WCNS) is presented to solve the full compressible Euler equations at all Mach numbers. To avoid the stringent acoustic CFL restriction for an explicit time discretization method, the pressure splitting methodology is introduced to split the Euler equations into nonstiff and stiff terms. The nonstiff and stiff terms are treated explicitly and implicitly, respectively, based on the high order IMEX Runge-Kutta time discretization method that is asymptotic preserving and asymptotically accurate in the zero Mach number limit. A fifth-order WCNS and fourth-order centered finite difference schemes with zero numerical viscosity are used for the spatial discretization. Numerical tests in one and two space dimensions are displayed to demonstrate the performance of the high order semi-implicit WCNS in compressible and incompressible regimes. Comparisons with the third-order explicit weighted essentially non-oscillatory (WENO) scheme are also made in order to better assess the proposed scheme.

Discrete maximum principle of a high order finite difference scheme for a generalized Allen–Cahn equation

We consider solving a generalized Allen-Cahn equation coupled with a passive convection for a given incompressible velocity field. The numerical scheme consists of the first order accurate stabilized implicit explicit time discretization and a fourth order accurate finite difference scheme, which is obtained from the finite difference formulation of the $Q^2$ spectral element method. We prove that the discrete maximum principle holds under suitable mesh size and time step constraints. The same result also applies to construct a bound-preserving scheme for any passive convection with an incompressible velocity field.

Kinetic discretization of one-dimensional nonlocal flow models

We show that one-dimensional nonlocal flow models in PDE form with Lighthill-Whitham-Richards flux supplemented with appropriate in- and out-flow terms can be spatially discretized with a finite volume scheme to obtain formally kinetic models with physically meaningful reaction graph structure. This allows the utilization of the theory of chemical reaction networks, as demonstrated here via the stability analysis of a flow model with circular topology. We further propose an explicit time discretization and a Courant-Friedrichs-Lewy condition ensuring many advantageous properties of the scheme. Additional characteristics, including monotonicity and the total variation diminishing property are also discussed.

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

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.

Dynamical System Related to Primal–Dual Splitting Projection Methods

We introduce a dynamical system to the problem of finding zeros of the sum of two maximally monotone operators. We investigate the existence, uniqueness and extendability of solutions to this dynamical system in a Hilbert space. We prove that the trajectories of the proposed dynamical system converge strongly to a primal–dual solution of the considered problem. Under explicit time discretization of the dynamical system we obtain the best approximation algorithm for solving coupled monotone inclusion problem.

Preconditioned Accelerated Gradient Descent Methods for Locally Lipschitz Smooth Objectives with Applications to the Solution of Nonlinear PDEs

We develop a theoretical foundation for the application of Nesterov’s accelerated gradient descent method (AGD) to the approximation of solutions of a wide class of partial differential equations (PDEs). This is achieved by proving the existence of an invariant set and exponential convergence rates when its preconditioned version (PAGD) is applied to minimize locally Lipschitz smooth, strongly convex objective functionals. We introduce a second-order ordinary differential equation (ODE) with a preconditioner built-in and show that PAGD is an explicit time-discretization of this ODE, which requires a natural time step restriction for energy stability. At the continuous time level, we show an exponential convergence of the ODE solution to its steady state using a simple energy argument. At the discrete level, assuming the aforementioned step size restriction, the existence of an invariant set is proved and a matching exponential rate of convergence of the PAGD scheme is derived by mimicking the energy argument and the convergence at the continuous level. Applications of the PAGD method to numerical PDEs are demonstrated with certain nonlinear elliptic PDEs using pseudo-spectral methods for spatial discretization, and several numerical experiments are conducted. The results confirm the global geometric and mesh size-independent convergence of the PAGD method, with an accelerated rate that is improved over the preconditioned gradient descent (PGD) method.

Contrast-independent partially explicit time discretizations for multiscale flow problems

Many multiscale problems have a high contrast, which is expressed as a very large ratio between the media properties. The contrast is known to introduce many challenges in the design of multiscale methods and domain decomposition approaches. These issues to some extent are analyzed in the design of spatial multiscale and domain decomposition approaches. However, some of these issues remain open for time dependent problems as the contrast affects the time scales, particularly, for explicit methods. For example, in parabolic equations, the time step is dt=H2/κmax, where κmax is the largest diffusivity. In this paper, we address this issue in the context of parabolic equation by designing a splitting algorithm. The proposed splitting algorithm treats dominant multiscale modes in the implicit fashion, while the rest in the explicit fashion. The contrast-independent stability of these algorithms requires a special multiscale space design, which is the main purpose of the paper. We show that with an appropriate choice of multiscale spaces we can achieve an unconditional stability with respect to the contrast. This could provide computational savings as the time step in explicit methods is adversely affected by the contrast. We discuss some theoretical aspects of the proposed algorithms. Numerical results are presented.

A hybrid Laplace‐Galerkin method for thermo‐hydro‐mechanical coupling in fluid saturated porous media: Application for borehole problems

AbstractA hybrid Laplace‐Galerkin method is proposed for modeling thermo‐hydro‐mechanical coupled problems of fluid saturated porous media in unsteady state regime. It consists of transforming the weak form equations into the Laplace space where they are solved by the Galerkin finite element method. Results in time space are then obtained by the Stehfest method of inverse Laplace transformation. This technique does not require any time stepping comparing to the traditional implicit or explicit time discretization method. Its exactitude is confirmed by comparing with existent analytical results as well as the one obtained by the implicit time stepping method.

Second Order Fully Semi-Lagrangian Discretizations of Advection-Diffusion-Reaction Systems

We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which is based on a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. The proposed method allows to use large time steps, while avoiding the solution of large linear systems, which would be required by implicit time discretization techniques. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. A novel extrapolation technique is proposed to enforce second-order accurate Dirichlet boundary conditions. We include a theoretical analysis of the scheme, along with numerical experiments which demonstrate the effectiveness of the proposed approach and its superior efficiency with respect to more conventional explicit and implicit time discretizations.

Hybrid Discontinuous Galerkin/Finite Volume Method with Subcell Resolution for Shocked Flows

A hybrid discontinuous Galerkin (DG)/finite volume (FV) method under the multistage boundary variation diminishing (BVD) reconstruction paradigm is developed for shocked inviscid and viscous flows. The methodology uses high-order unlimited polynomials and the THINC (which stands for tangent of hyperbola for interface capturing) function as the candidate reconstruction functions to perform FV computation on embedded subcells. The reconstruction scheme on the FV subcells can be designed in a way to keep the resulting approach as compact as the underlying DG method. The reconstruction information based on the modified BVD selection algorithm has been applied as an indicating strategy so that no additional discontinuity indicator is required for the hybrid DG/FV computation. The designed approach has been investigated through the coupling with the explicit and implicit time discretization scheme for both inviscid and viscous flows. Numerical results demonstrate the high-fidelity characteristics of the method by using a much coarser grid to capture both the discontinuity and the smooth flow structure with high resolution.

High Order Semi-Implicit WCNS for 1D Isentropic and Full Euler Equations

This paper designs a high order semi-implicit weighted compact nonlinear scheme (WCNS) to solve 1D isentropic and full Euler equations. The Euler equations are split into stiff and non-stiff terms that are solved by the implicit and explicit time discretization method, respectively, in order to improve the computational efficiency for low Mach flows. The fifth-order WCNS is applied for the spatial discretization. Several numerical examples are given to demonstrate the performance of the designed method.

Primal Dual Methods for Wasserstein Gradient Flows

Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank–Nicolson-type method we introduce. Next, we use the Benamou–Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints. Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators (Yan in J Sci Comput 1–20, 2018). By leveraging the PDEs’ underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations’ strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method.

Implicit and explicit discrete‐time realizations of homogeneous differentiators

AbstractThis article presents explicit and implicit discrete‐time realizations for a class of homogeneous sliding mode‐based differentiators. The proposed approach relies on the method of exact discretization for linear systems with a zero‐order holder. The two discrete‐time schemes present the homogeneity property and, after a finite time, the accuracy of its continuous‐time counterpart, even in the presence of bounded noise. In comparison to the explicit realization, which makes it possible to determine the state of the system later from the state at present, for this case, the implicit method requires finding a solution by solving generalized equations that involve the current state of the system and two support variables. Therefore, this document proves that the polynomial's unique positive root defines the required solution as part of the main results. Hence, it is possible to introduce a non‐anticipative method to implement the implicit discrete‐time realization, including an appropriate root‐finding method for the polynomial. Finally, the simulation results include comparisons between the proposed implicit and explicit discrete methods with other existing schemes. Numerical studies clearly show that the implicit method supersedes the explicit one, consistent with the implicit and explicit time discretization of other continuous‐time algorithms.

Correction to: An implicit–explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions

Correction to: An implicit–explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions

Implementation of the fractional-step, artificial compressibility with pressure projection (FSAC-PP) method into openfoam for unsteady flows

Commercial and open-source CFD solvers rely mostly on incompressible approximate projection methods to overcome the pressure-velocity decoupling, such as the SIMPLE (Patankar, 1980) or PISO (Issa, 1986) algorithm. Incompressible methods based on the Artificial Compressibility method (Chorin, 1967) lack a mechanism to evolve in time and need to be supplemented by a real time derivative through the dual time scheme. The current study investigates the implementation of the explicit dual time discretization of the Artificial Compressibility method into OpenFOAM and extends on that by applying the dual time scheme to the incompressible FSAC-PP method (Könözsy, 2012). Applied to the Couette 2D flow at Re=100 and Re=1000, results show that for both methods accurate time evolutions of the velocity profiles are presented, where the FSAC-PP methods seemingly produces smoother profiles compared to the AC method, especially during the start-up of the simulation.

Finite element analysis of coupled phase-field and thermoelasticity equations at large strains for martensitic phase transformations based on implicit and explicit time discretization schemes

In this paper, a staggered, nonlinear finite element procedure is developed to solve the large-strain based coupled system of time dependent Ginzburg-Landau (GL) and thermoelasticity equations for phase transformations at the nanoscale. Geometrical nonlinearities are included based on the total Lagrangian description where the total deformation gradient is defined as the multiplicative decomposition of elastic, transformational and thermal deformation gradient tensors. The principle of virtual work is utilized to obtain the integral form of the Lagrangian equation of motion whose discretized form is solved using the iterative methods which gives the thermal and transformational deformation gradient tensors. Next, the elastic deformation gradient tensor and the first Piola-Kirchhoff stress are calculated and substituted in the GL equation which gives the phase order parameter. The weighted residual method is used to derive the corresponding finite element form which allows for the phase-dependent surface energy BCs. Explicit and different implicit methods belonging to the generalized trapezoidal family are used for time discretization of the GL equation. Various examples of phase transformations are simulated and discussed. The effect of different time integration schemes are also investigated. The current study allows for a proper modeling of various phase field problems at large and small strains.