Implicit-explicit Time Discretization Research Articles

Local Discontinuous Galerkin Methods with Multistep Implicit–Explicit Time Discretization for Nonlinear Schrödinger Equations

Local Discontinuous Galerkin Methods with Multistep Implicit–Explicit Time Discretization for Nonlinear Schrödinger Equations

Implicit-Explicit Time Discretization for Oseen’s Equation at High Reynolds Number with Application to Fractional Step Methods

Implicit-Explicit Time Discretization for Oseen’s Equation at High Reynolds Number with Application to Fractional Step Methods

An implicit–explicit time discretization for elastic wave propagation problems in plates

AbstractWe propose a new implicit–explicit scheme to address the challenge of modeling wave propagation within thin structures using the time‐domain finite element method. Compared to standard explicit schemes, our approach renders a time marching algorithm with a time step independent of the plate thickness and its associated discretization parameters (mesh step and order of approximation). Relying on the standard three dimensional elastodynamics equations, our strategy can be applied to any type of material, either isotropic or anisotropic, with or without discontinuities in the thickness direction. Upon the assumption of an extruded mesh of the plate‐like geometry, we show that the linear system to be solved at each time step is partially lumped thus efficiently treated. We provide numerical evidence of an adequate convergence behavior, similar to a reference solution obtained using the well‐known leapfrog scheme. Further numerical investigations show significant speed up factors compared to the same reference scheme, proving the efficiency of our approach for the configurations of interest.

A finite-volume scheme for modeling compressible magnetohydrodynamic flows at low Mach numbers in stellar interiors

Fully compressible magnetohydrodynamic (MHD) simulations are a fundamental tool for investigating the role of dynamo amplification in the generation of magnetic fields in deep convective layers of stars. The flows that arise in such environments are characterized by low (sonic) Mach numbers (ℳson ≲ 10−2). In these regimes, conventional MHD codes typically show excessive dissipation and tend to be inefficient as the Courant–Friedrichs–Lewy (CFL) constraint on the time step becomes too strict. In this work we present a new method for efficiently simulating MHD flows at low Mach numbers in a space-dependent gravitational potential while still retaining all effects of compressibility. The proposed scheme is implemented in the finite-volume SEVEN-LEAGUE HYDRO (SLH) code, and it makes use of a low-Mach version of the five-wave Harten–Lax–van Leer discontinuities (HLLD) solver to reduce numerical dissipation, an implicit–explicit time discretization technique based on Strang splitting to overcome the overly strict CFL constraint, and a well-balancing method that dramatically reduces the magnitude of spatial discretization errors in strongly stratified setups. The solenoidal constraint on the magnetic field is enforced by using a constrained transport method on a staggered grid. We carry out five verification tests, including the simulation of a small-scale dynamo in a star-like environment at ℳson ~ 10−3. We demonstrate that the proposed scheme can be used to accurately simulate compressible MHD flows in regimes of low Mach numbers and strongly stratified setups even with moderately coarse grids.

A POD-RBF-FD scheme for simulating chemotaxis models on surfaces

The main aim of this paper is to develop a new framework of a meshless approximation for solving numerically three nonlinear partial differential equations in biology, i.e., the chemotaxis models defined on the smooth, closed manifolds embedded in R3. The radial basis function-generated finite difference scheme is considered to deal with the spatial variables, which depends only on the location of nodes and the value of normal vector at each point per spherical cap. The robust artificial hyperviscosity formulation is derived for each model, which has been used for preventing the spurious growth modes in the numerical solution. An implicit–explicit time discretization is employed to deal with the time variable. The resulting fully discrete scheme is solved via the biconjugate gradient stabilized method with a zero-fill incomplete lower upper preconditioner per time step, where a positivity-preserving filter is used to prevent the negative sign of the cell density variable. Besides, to reduce the used central processor unit (CPU) time, the proper orthogonal decomposition is considered for constructing a set of new orthogonal basis vectors based on the singular value decomposition. The developed numerical method is called a proper orthogonal decomposition-radial basis function-generated finite difference (POD-RBF-FD) scheme. Finally, the ability of the proposed method is investigated by simulation results showing the blowing-up, pattern formulation (perforated stripe) and aggregations of bacteria on some surfaces.

An entropy stable scheme for the non-linear Boltzmann equation

The kinetic equations govern the behavior of gaseous flows, compressibility, turbulence, reactions with internal energy exchange, that form the essence of numerous physical processes. Despite their wide applicability, their six-dimensional (seven including time) nature presents a huge computational challenge. With the advent of the modern high performance computing systems and few recent advances in numerical methods, it is now possible to numerically study the behavior of these systems. However, to understand the instability of rich molecular processes and find a structure in the apparent chaos, a scheme that is efficient in multi-dimensions, exhibits high parallel-efficiency, and foremost produces an entropy solution as per the theory of solutions of hyperbolic systems may prove useful. In this work, first, we construct an entropy stable flux for non-linear inhomogeneous (full) Boltzmann equation. Second, to ensure geometrical flexibility, we couple the scheme with a class of high order discontinuous Galerkin discretization (Jaiswal 2019 [41]) which satisfy summation-by-part (SBP) discretely. Third, utilizing SBP, we prove that the resulting semi-discrete scheme is locally and globally conservative in flat spaces; preserves the entropy-decay (H-theorem) property; efficient and simple; and therefore suitable for dealing with highly complex non-smooth flow problems. Fourth, we show that the fully-discrete kinetic scheme, utilizing an implicit-explicit time-discretization, in the limit of vanishing Knudsen number, becomes an entropy-stable explicit scheme applied to the Euler system. Fifth, we carry out a series of verification tests to illustrate the stability, accuracy, and conservation properties of the proposed method. These tests involve over 85 million degrees of freedom and over 50 billion operations per time-step.

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.

Non-linear Boltzmann equation on hybrid-unstructured non-conforming multi-domains

Adaptivity is crucial for addressing practical challenges of the next decades. In this regard, recent surveys [117] highlight that it continues to be a major bottleneck in computational fluid dynamics workflow. We introduce mixed non-conforming discontinuous Galerkin discretization for the full Boltzmann equation in 2D/3D. These schemes have been designed for efficiency — motivated in part by spectral and isogeometric weighted collocation methods — and retain an optimal O(p+1) convergence for a p order approximation for non-linear kinetic systems on non-orthogonal grids. In this setting, it is possible to analyze highly complex problems of industrial strength i.e., structured, unstructured, mixed, irregular, multi-domain (multi-block) adaptive geometries at massively parallel scale (ten thousand cores or beyond). Mixed domains permit flexible mesh generation, whereas local nature of discontinuous Galerkin permits construction of adaptive numerical schemes that scale well. To address flows in mixing regime (low/high rarefaction), we couple the scheme with an asymptotic preserving implicit-explicit time discretization. These schemes are iteration free and applicable for a wide range of rarefied flows from free-molecular to continuum. To ensure stability in presence of shocks, we describe a method of constructing limiters on non-conforming grids. Finally, we show that the computational overhead for solving kinetic equations on non-conforming structured/unstructured domains is negligible relative to conforming domains. So, there is no reason to not prefer non-conforming unstructured domains.

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

Energy stable finite element method for an electrohydrodynamic model with variable density

In this paper, we present two linear time-marching methods for an electrohydrodynamic model of charge transport with variable density dielectric fluid. The model is a highly coupled nonlinear system and an energy dissipation law of the system is discovered. The main feature of the proposed methods is that energy stability is preserved at the fully discrete level by using suitable reformulations of the continuous equations and implicit-explicit time discretization schemes. The mixed finite element method is employed for the spatial discretization. Numerical experiments are carried out to validate the convergence rates and the energy stability of the schemes. To further demonstrate the robustness of the proposed schemes, the effects of injected charges for Rayleigh-Taylor instability and the electro-convection phenomenon between two concentric half-cylinders are simulated. Results show that the injected charges can accelerate the evolutions of Rayleigh-Taylor instability and the number of vortices increases in the electro-convective flow when the inner radius increases.

A splitting method adapted to the simulation of mixed flows in pipes with a compressible two-layer model

The numerical resolution of the Compressible Two-Layer model proposed in [27] is addressed in this work with the aim of simulating mixed flows and entrapped air pockets in pipes. This five-equation model provides a unified two-phase description of such flows which involve transitions between stratified regimes (air–water herein) and pressurized or dry regimes (pipe full of water or air). In particular, strong interactions between both phases and entrapped air pockets are accounted for. At the discrete level, the coexistence of slow gravity waves in the stratified regime with fast acoustic waves in the pressurized regime is difficult to approximate. Furthermore, the two-phase description requires to deal with vanishing phases in pressurized and dry regimes. In that context, a robust splitting method combined with an implicit-explicit time discretization is derived. The overall strategy relies on the fast pressure relaxation in addition to a mimetic approach with the shallow water equations for the slow dynamics of the water phase. It results in a three-step scheme which ensures the positivity of heights and densities under a CFL condition based on the celerity of material and gravity waves. In that framework, an implicit relaxation-like approach provides stabilization terms which are activated according to the flow regime. Numerical experiments are performed beginning with a Riemann problem for the convective part. The overall approach is then assessed considering relevant mixed flow configurations involving regime transitions, vanishing phases and entrapped air pockets.

An RBF–FD method for pricing American options under jump–diffusion models

American option prices under jump–diffusion models are determined as solutions to partial integro-differential equations (PIDE). In this paper a new combination of a time and spatial discretization applied to a linear complementary formulation (LCP) of the free boundary PIDE is proposed. First a coordinate stretching transformation is performed to the asset price so that the computation of the prices can be focused on regions of real interest instead of on the whole solution domain. An implicit–explicit time discretization applied to the reformulated LCP on a uniform temporal grid is followed by a spatial discretization to get a fully discrete system. The radial basis function (RBF) finite difference method is a local method resulting in a sparse linear system in contrast to global RBF-methods which lead to ill-conditioned dense matrix systems. For the corresponding European option we prove consistency, stability and second-order convergence in a discrete L2-norm. We derive mild conditions for the model parameters under which these results hold. Numerical experiments are performed with European and American options, and a comparison with numerical results available in the literature illustrates the accuracy and efficiency of the proposed algorithm.

Numerical simulation of a compressible two-layer model: A first attempt with an implicit–explicit splitting scheme

This paper is devoted to the numerical simulation of the compressible two-layer model developed in Demay and Hérard (2017). The latter is a hyperbolic two-fluid two-pressure model dedicated to gas–liquid flows in pipes, especially stratified air–water flows. Using explicit schemes, one obtains a CFL condition based on the celerity of (fast) acoustic waves which typically brings large numerical diffusivity for the (slow) material waves and small time steps. In order to overcome these drawbacks, the proposed scheme involves an operator splitting and an implicit–explicit time discretization. Thus, the full system is split into two hyperbolic sub-systems. The first one deals with the transport equation on the liquid height using an explicit scheme and upwind fluxes. The second one deals with the averaged mass and momentum conservation equations of both phases using an implicit scheme which handles the propagation of acoustic waves. At last, the positivity of heights and densities is ensured under a CFL condition which involves material velocities. Numerical experiments are performed using acoustic as well as material time steps. Adding the Rusanov scheme for comparison, the best accuracy is obtained with the proposed scheme used with acoustic time steps. Focusing on material waves of the convective system, the efficiency of the latter is improved when using material steps. However, considering the whole system with relaxation source terms, an efficient approximation of slow dynamics, typically a gravity driven flow, is still challenging.

Local Discontinuous Galerkin Method with Implicit–Explicit Time Marching for Incompressible Miscible Displacement Problem in Porous Media

In this paper, we shall present two fully-discrete local discontinuous Galerkin methods, coupled with multi-step implicit–explicit time discretization up to second order, for solving the two-dimensional incompressible miscible displacement problem. To avoid the solving of nonlinear algebraic systems, the extrapolation linearization is adopted to diffusion–dispersion tensor. Under weak temporal-spatial conditions, the optimal error estimates in $$L^{\infty }(L^{2})$$ norm for both concentration and velocity are derived. Numerical experiments are also given to demonstrate the theoretical results.

Parameter estimation in IMEX-trigonometrically fitted methods for the numerical solution of reaction–diffusion problems

In this paper, an adapted numerical scheme for reaction–diffusion problems generating periodic wavefronts is introduced. Adapted numerical methods for such evolutionary problems are specially tuned to follow prescribed qualitative behaviors of the solutions, making the numerical scheme more accurate and efficient as compared with traditional schemes already known in the literature. Adaptation through the so-called exponential fitting technique leads to methods whose coefficients depend on unknown parameters related to the dynamics and aimed to be numerically computed. Here we propose a strategy for a cheap and accurate estimation of such parameters, which consists essentially in minimizing the leading term of the local truncation error whose expression is provided in a rigorous accuracy analysis. In particular, the presented estimation technique has been applied to a numerical scheme based on combining an adapted finite difference discretization in space with an implicit–explicit time discretization. Numerical experiments confirming the effectiveness of the approach are also provided.

Efficient Stochastic Asymptotic-Preserving Implicit-Explicit Methods for Transport Equations with Diffusive Scalings and Random Inputs

For linear transport and radiative heat transfer equations with random inputs, we develop new generalized polynomial chaos based asymptotic-preserving stochastic Galerkin schemes that allow efficient computation for the problems that contain both uncertainties and multiple scales. Compared with previous methods for these problems, our new method uses the implicit-explicit time discretization to gain higher order accuracy, and by using a modified diffusion operator based penalty method, a more relaxed stability condition---a hyperbolic, rather than parabolic, CFL stability condition---is achieved in the case of a small mean free path in the diffusive regime. The stochastic asymptotic-preserving property of these methods will be shown asymptotically and demonstrated numerically, along with a computational cost comparison with previous methods.

A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations

A fourth-order implicit-explicit time-discretization scheme based on the exponential time differencing approach with a fourth-order compact scheme in space is proposed for space fractional nonlinear Schrodinger equations. The stability and convergence of the compact scheme are discussed. It is shown that the compact scheme is fourth-order convergent in space and in time. Numerical experiments are performed on single and coupled systems of two and four fractional nonlinear Schrodinger equations. The results demonstrate accuracy, efficiency, and reliability of the scheme. A linearly implicit conservative method with the fourth-order compact scheme in space is also considered and used on the system of space fractional nonlinear Schrodinger equations.

Implicit–explicit time discretization coupled with finite element methods for delayed predator–prey competition reaction–diffusion system

The main purpose of this paper is to develop the two-step implicit–explicit(IMEX) time discretization coupled with finite element methods for solving delayed predator–prey competition reaction–diffusion system. Finite element methods are used to discretize the space variables, both IMEX two-step one-leg methods and IMEX linear two-step methods are considered in time discretization, where the nonlinear reaction part is discretized explicitly and the diffusion part is discretized implicitly. The L2-norm error estimates with handling the breaking points are derived. These methods only require solving two independent linear systems with the same coefficient matrix at different time, and avoid solving large-scale systems of nonlinear algebraic equations, thus the resulting schemes are efficient. Some numerical experiments are presented to confirm the theoretical results.

Unsaturated subsurface flow with surface water and nonlinear in- and outflow conditions

We analytically and numerically analyze groundwater flow in a homogeneous soil described by the Richards equation, coupled to surface water represented by a set of ordinary differential equations (ODEs) on parts of the domain boundary, and with nonlinear outflow conditions of Signorini's type. The coupling of the partial differential equation (PDE) and the ODE's is given by nonlinear Robin boundary conditions. This paper provides two major new contributions regarding these infiltration conditions. First, an existence result for the continuous coupled problem is established with the help of a regularization technique. Second, we analyze and validate a solver-friendly discretization of the coupled problem based on an implicit–explicit time discretization and on finite elements in space. The discretized PDE leads to convex spatial minimization problems which can be solved efficiently by monotone multigrid. Numerical experiments are provided using the DUNE numerics framework.

Design and Analysis of a Lightweight Parallel Adaptive Scheme for the Solution of the Monodomain Equation

Numerical simulation of the nonlinear reaction-diffusion equations in computational electrocardiology requires locally high spatial resolution to capture the multiscale effects related to the electrical activation of the heart accurately, namely the strongly varying transmembrane potential. Here, we propose a novel lightweight adaptive algorithm which aims at combining the plainness of structured meshes with the resolving capabilities of unstructered adaptive meshes. Our “patchwise adaptive” approach is based on locally structured mesh hierarchies which are glued along their interfaces by a nonconforming mortar element discretization. To further increase the overall efficiency, we keep the spatial meshes constant over suitable time windows in which error indicators are accumulated. This approach facilitates strongly varying mesh sizes in neighboring patches as well as in consecutive time steps. For the transfer of the dynamic variables between different spatial approximation spaces we compare the $L^2$-projection and a local approximation. Finally, since an implicit-explicit time discretization is employed for stability reasons, we derive a spatial preconditioner which is tailored to the special structure of the patchwise adaptive meshes. We analyze the (parallel) performance and scalability of the resulting method by several examples from computational electrocardiology of different sizes. Additionally, we compare our method to a standard adaptive refinement strategy using unstructured meshes. As it turns out, our novel adaptive scheme provides a very good balance between reduction in degrees of freedom and overall (parallel) efficiency.