Implicit-explicit Scheme Research Articles

Surface Flux Transport Modeling Using Physics-informed Neural Networks

Studying the magnetic field properties on the solar surface is crucial for understanding the solar and heliospheric activities, which in turn shape space weather in the solar system. Surface flux transport (SFT) modeling helps us to simulate and analyze the transport and evolution of magnetic flux on the solar surface, providing valuable insights into the mechanisms responsible for solar activity. In this work, we demonstrate the use of machine learning techniques in solving magnetic flux transport, making it accurate. We have developed a novel physics-informed neural network (PINN)-based model to study the evolution of bipolar magnetic regions using SFT in one-dimensional azimuthally averaged and also in two dimensions. We demonstrate the efficiency and computational feasibility of our PINN-based model by comparing its performance and accuracy with that of a numerical model implemented using the Runge–Kutta implicit–explicit scheme. The mesh-independent PINN method can be used to reproduce the observed polar magnetic field with better flux conservation. This advancement is important for accurately reproducing observed polar magnetic fields, thereby providing insights into the strength of future solar cycles. This work paves the way for more efficient and accurate simulations of solar magnetic flux transport and showcases the applicability of PINNs in solving advection–diffusion equations with a particular focus on heliophysics.

High-order accurate implicit-explicit time-stepping schemes for wave equations on overset grids

New implicit and implicit-explicit time-stepping methods for the wave equation in second-order form are described with application to two and three-dimensional problems discretized on overset grids. The implicit schemes are single step, three levels in time, and based on the modified equation approach. Second and fourth-order accurate schemes are developed and they incorporate upwind dissipation for stability on overset grids. The fully implicit schemes are useful for certain applications such as the WaveHoltz algorithm for solving Helmholtz problems where very large time-steps are desired. Some wave propagation problems are geometrically stiff due to localized regions of small grid cells, such as grids needed to resolve fine geometric features, and for these situations the implicit time-stepping scheme is combined with an explicit scheme: the implicit scheme is used for component grids containing small cells while the explicit scheme is used on the other grids such as background Cartesian grids. The resulting partitioned implicit-explicit scheme can be many times faster than using an explicit scheme everywhere. The accuracy and stability of the schemes are studied through analysis and numerical computations.

Spatially mixed implicit–explicit schemes in hydro-mechanically coupled soil dynamics

Mixed implicit–explicit (IMEX) schemes combine elements of both implicit and explicit time integration methods. This allows to efficiently handle boundary value problems with severe differences in stiffness, such as often encountered in soil dynamics. A spatially mixed IMEX scheme, in which different parts of the computational domain may be treated implicitly or explicitly depending on their behaviour, is discussed in this work. Domain decomposition by means of a mortar contact discretisation is used for this purpose and multiple systems of equations are solved. The merits of the scheme in solving problems of soil dynamics are demonstrated, for which the governing equations are extended to hydro-mechanical coupling.

Unconditional error analysis of weighted implicit-explicit virtual element method for nonlinear neutral delay-reaction–diffusion equation

In this paper, we develop a virtual element method in space for the nonlinear neutral delay-reaction–diffusion equation, while a weighted implicit-explicit scheme is utilized in time. The nonlinear term is adopted by using the Newton linearized method. The calculation efficiency is improved using an implicit scheme to analyze linear terms and an explicit scheme to deal with nonlinear terms. Then, the time–space error splitting technique and G-stability are creatively combined to rigorously analyze the unconditionally optimal convergence results of the numerical scheme without any restrictions on the mesh ratio. Finally, numerical examples on a set of polygonal meshes are provided to confirm the theoretical results. In particular, when the weighted parameter is taken θ=12 (or θ=1), the method degenerates into the Crank–Nicolson (or second-order backward differential formula (BDF2)) scheme.

Unified Leapfrog CDI-FDTD Methods With Weakly Conditional Stability and Hybrid Implicit-Explicit Schemes

Unified Leapfrog CDI-FDTD Methods With Weakly Conditional Stability and Hybrid Implicit-Explicit Schemes

Energy-conserving discontinuous Galerkin methods for the Vlasov-Ampère system with Dougherty-Fokker-Planck collision operator

This paper develops energy-preserving discontinuous Galerkin (DG) methods for the Vlasov-Ampère (VA) system coupled with the Dougherty-Fokker-Planck (DFP) collision operator. While the classical VA system has been extensively studied, the inclusion of the collision operator introduces new challenges in conserving the total energy of the system. To address this, we design two energy-conserving temporal discretization methods: a second-order explicit scheme and a second-order implicit-explicit (IMEX) scheme. These schemes are coupled with the DG method, specifically using the local DG (LDG) method for the DFP part. We prove that the fully discrete schemes conserve the total particle number and total energy of the VA-DFP system at the fully discrete level. We further establish the L2 stability of the fully discrete explicit scheme. Numerical experiments are conducted to assess the accuracy, conservation property, and performance of the proposed schemes.

Explicit–Implicit Schemes for Calculating Dynamics of Elastoviscoplastic Media with Softening

Explicit–Implicit Schemes for Calculating Dynamics of Elastoviscoplastic Media with Softening

An explicit–implicit Generalized Finite Difference scheme for a parabolic–elliptic density-suppressed motility system

In this work, a Generalized Finite Difference (GFD) scheme is presented for effectively computing the numerical solution of a parabolic–elliptic system modeling a bacterial strain with density-suppressed motility. The GFD method is a meshless method known for its simplicity for solving non-linear boundary value problems over irregular geometries. The paper first introduces the basic elements of the GFD method, and then an explicit–implicit scheme is derived. The convergence of the method is proven under a bound for the time step, and an algorithm is provided for its computational implementation. Finally, some examples are considered comparing the results obtained with a regular mesh and an irregular cloud of points.

Computational modeling of cracking in cortical bone microstructure using the mesh fragmentation technique

The cortical bone is a hierarchical composite material that, at the microscale, is segmented in an interstitial matrix, cement line, osteons, and Haversian canals. The cracking of the structure at this scale directly influences the macro behavior, and, in this context, the cement line has a protagonist role. In this sense, this work aims to simulate the crack initiation and propagation processes via cortical bone microstructure modeling with a two-dimensional mesh fragmentation technique that captures the mechanical relevance of its constituents. In this approach, high aspect ratio elements are inserted between the regular constant strain triangle finite elements to define potential crack paths a priori. The crack behavior is described using a composed damage model with two scalar damage variables, which is integrated by an implicit-explicit (Impl-Ex) scheme to avoid convergence problems usually found in numerical simulations involving multiple cracks. The approach’s capability of modeling the failure process in cortical bone microstructure is investigated by simulating four conceptual problems and one example based on a digital image of an experimental test. The results obtained in terms of crack pattern and failure mechanisms agree with those described in the literature, demonstrating that the numerical tool is promising to simulate the complex failure mechanisms in cortical bone, considering the properties of its distinct phases.

WENO scheme on characteristics for the equilibrium dispersive model of chromatography with generalized Langmuir isotherms

Column chromatography is a laboratory and industrial technique used to separate different substances mixed in a solution. Mathematically, it can be modeled using non-linear partial differential equations whose main ingredients are the adsorption isotherms, which are non-linear functions modeling the affinity between the different substances in the solution and the solid stationary phase filling the column. The goal of this work is twofold. Firstly, we aim to extend the techniques of Donat, Guerrero and Mulet (2018) [3] to other adsorption isotherms. In particular, we propose a family of generalized Langmuir-type isotherms and prove that the correspondence between the concentrations of solutes in the liquid phase (the primitive variables) and the conserved variables is well defined and admits a global smooth inverse that can be computed numerically. Secondly, to establish the well-posedness of the mathematical model, we study the eigenstructure of the Jacobian of the mentioned correspondence and use this characteristic information to get oscillation-free sharp interfaces on the numerical approximate solutions. To do so, we determine the structure of the Jacobian matrix of the system and use it to deduce its eigenstructure. We combine the use of characteristic-based numerical fluxes with a second-order implicit-explicit scheme proposed in the cited reference and perform some numerical experiments with Tóth's adsorption isotherms to demonstrate that the characteristic-based schemes produce accurate numerical solutions with no oscillations, even when steep gradients appear in the solutions.

Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review

Hyperbolic systems with stiff relaxation constitute a wide class of evolutionary partial differential equations which describe several physical phenomena, ranging from gas dynamics to kinetic theory, from semiconductor modeling to traffic flow. Peculiar features of such systems is the presence of a small parameter that determines the smallest time scale of the system. As such parameter vanishes, the system relaxes to a different one with a smaller number of equations, and sometime of a different mathematical nature. The numerical solution of such systems may present some challenges, in particular if one is interested in capturing all regimes with the same numerical method, including the one in which the small parameter vanishes (relaxed system). The design, analysis and application of numerical schemes which are robust enough to solve this class of systems for arbitrary value of the small parameter is the subject of the current paper. We start presenting different classes of hyperbolic systems with relaxation, illustrate the properties of implicit–explicit (IMEX) Runge–Kutta schemes which are adopted for the construction of efficient methods for the numerical solution of the systems, and then illustrate how to apply IMEX schemes for the construction of asymptotic preserving schemes, i.e. scheme which correctly capture the behavior of the systems even when the relaxation parameter vanishes.

A Fast Second-order Solver for Stiff Multifluid Dust and Gas Hydrodynamics

We present MDIRK: a multifluid second-order diagonally implicit Runge–Kutta method to study momentum transfer between gas and an arbitrary number (N) of dust species. The method integrates the equations of hydrodynamics with an implicit–explicit scheme and solves the stiff source term in the momentum equation with a diagonally implicit, asymptotically stable Runge–Kutta method (DIRK). In particular, DIRK admits a simple analytical solution that can be evaluated with O(N) operations, instead of standard matrix inversion, which is O(N)3 . Therefore, the analytical solution significantly reduces the computational cost of the multifluid method, making it suitable for studying the dynamics of systems with particle-size distributions. We demonstrate that the method conserves momentum to machine precision and converges to the correct equilibrium solution with constant external acceleration. To validate our numerical method we present a series of simple hydrodynamic tests, including damping of sound waves, dusty shocks, a multifluid dusty Jeans instability, and a steady-state gas–dust drift calculation. The simplicity of MDIRK lays the groundwork to build fast high-order, asymptotically stable multifluid methods.

A finite difference explicit-implicit scheme for fractal heat and mass transportation of Williamson nanofluid flow in quantum calculus

This study introduces a novel and versatile fractal finite difference scheme designed to address unsteady flow challenges in quantum calculus over flat and oscillatory sheets. Motivated by the need to advance our understanding of nanofluid dynamics, particularly in heat and mass transportation, this research aims to bridge existing gaps and contribute to improving computational methodologies. Our approach involves a two-stage scheme incorporating explicit and implicit methods, focusing on q-derivatives for spatial terms within the mathematical model. The system encompasses the Navier-Stokes equation, energy equation, and concentration equation, offering a comprehensive framework for analyzing mixed convective Williamson nanofluid flow under the influence of chemical reactions. The convergence condition for the time-dependent partial differential equations system, particularly the scalar convection-diffusion equation, is presented. Stability requirements for our proposed fractal scheme are also delineated. Notably, our study explores the implications of the Weisenberg parameter on the velocity profile, revealing a decline that underscores its influence on nanofluid behavior. Our fractal finite difference scheme demonstrates a substantially faster convergence rate than the fractal Crank-Nicolson scheme. This outcome emphasizes the computational efficiency and efficacy of our proposed technique. In a broader context, our work contributes to the field by leveraging fractal geometry and quantum calculus to enhance our understanding of nanofluid dynamics. These tools present exciting opportunities for groundbreaking applications and technological advancements across various industries. As we delve into these research areas, our work sets the stage for exploration, providing a framework with great potential for transformative progress in nanofluid dynamics.

Accuracy Analysis for Explicit-Implicit Finite Volume Schemes on Cut Cell Meshes

The solution of time-dependent hyperbolic conservation laws on cut cell meshes causes the small cell problem: standard schemes are not stable on the arbitrarily small cut cells if an explicit time stepping scheme is used and the time step size is chosen based on the size of the background cells. In May and Berger (J Sci Comput 71: 919–943, 2017), the mixed explicit-implicit approach in general and MUSCL-Trap (monotonic upwind scheme for conservation laws and trapezoidal scheme) in particular have been introduced to solve this problem by using implicit time stepping on the cut cells. Theoretical and numerical results have indicated that this might lead to a loss in accuracy when switching between the explicit and implicit time stepping. In this contribution, we examine this in more detail and will prove in one dimension that the specific combination MUSCL-Trap of an explicit second-order and an implicit second-order scheme results in a fully second-order mixed scheme. As this result is unlikely to hold in two dimensions, we also introduce two new versions of mixed explicit-implicit schemes based on exchanging the explicit scheme. We present numerical tests in two dimensions where we compare the new versions with the original MUSCL-Trap scheme.

A fast time integral finite difference method for a space-time fractional FitzHugh-Nagumo monodomain model in irregular domains

This work aims at proposing a fast time integral (FTI) method for the space-time fractional Fitzhugh-Nagumo (FHN) monodomain model in irregular domains, which is commonly used to characterize the transmembrane potential of the heart. In order to reduce the storage and the algorithm complexity due to the geometric configuration of the heart, the derivation of the sum of the exponentials (cf. [1]) in FTI method is improved by the integral transformation, integral truncation and Gauss-Legendre quadrature. Such strategy is applied to approximate the Caputo fractional derivative. The number of the exponentials, reduced in the FTI method, is related to the calculation efficiency. The CPU time of the FHN monodomain model using FTI method is reduced by an order of magnitude. Moreover, a second-order discrete method to the Riesz space fractional derivative adopts for the spatial discretization, then an implicit-explicit scheme is derived for the nonlinear FHN model under the finite difference method. Numerical results are reported to demonstrate the convergence behaviors, robustness and high efficiency of the proposed method.

A splitting discontinuous Galerkin projection method for the magneto-hydrodynamic equations

This paper is focused on the numerical approximations of the incompressible magneto-hydrodynamic equations. Our interest is to develop a novel decoupled, linear and unconditional energy stable fully-discrete scheme, which is achieved by the interior penalty discontinuous Galerkin (DG) method for spatial discretization, the stabilizing strategy and implicit-explicit (IMEX) scheme used to handle the nonlinear coupling terms, and a rotational pressure-correction method for the Navier-Stokes equations. We prove the unique solvability, unconditional energy stability and optimal error estimates of the proposed scheme rigorously. We further present several numerical examples to demonstrate the accuracy, stability, and efficiency of the proposed scheme numerically.

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.

Explicit–implicit schemes for non-isothermal filtration problem: Single-temperature model

In this paper, we consider a non-isothermal filtration model for a two-phase immiscible incompressible fluid, for the numerical implementation of which an explicit–implicit algorithm is proposed. The problem is presented in a mixed generalized formulation, and an explicit–implicit method of the IMPES type is used to calculate the pressure, full velocity and water saturation, and an explicit–implicit scheme is built to calculate the temperature and heat flux with explicit consideration of convective heat transfer. At the beginning of the article, the mechanism of temperature homogenization of according to temperature heterogeneous system is considered. In computational experiments, special attention is paid to the comparison of the non-isothermal model with the isothermal one, including the discussion of fingering instability.

Parallel-in-time integration of the shallow water equations on the rotating sphere using Parareal and MGRIT

Despite the growing interest in parallel-in-time methods as an approach to accelerate numerical simulations in atmospheric modeling, improving their stability and convergence remains a substantial challenge for their application to operational models. In this work, we study the temporal parallelization of the shallow water equations on the rotating sphere combined with time-stepping schemes commonly used in atmospheric modeling due to their stability properties, namely an Eulerian implicit-explicit (IMEX) method and a semi-Lagrangian semi-implicit method (SL-SI-SETTLS). The main goal is to investigate the performance of parallel-in-time methods, namely Parareal and Multigrid Reduction in Time (MGRIT) when these well-established schemes are used on the coarse discretization levels and provide insights on how they can be improved for better performance. We begin by performing an analytical stability study of Parareal and MGRIT applied to a linearized ordinary differential equation depending on some temporal parallelization parameters, including the choice of a coarse scheme. Next, we perform numerical simulations of two standard tests in atmospheric modeling to evaluate the stability, convergence, and speedup provided by the parallel-in-time methods compared to a fine reference solution computed serially. We also conduct a detailed investigation on the influence of artificial viscosity and hyperviscosity approaches, applied on the coarse discretization levels, on the performance of the temporal parallelization. Both the analytical stability study and the numerical simulations indicate a poorer stability behavior when SL-SI-SETTLS is used on the coarse levels, compared to the IMEX scheme. With the IMEX scheme, a better trade-off between convergence, stability, and speedup compared to serial simulations can be obtained under proper parameters and artificial viscosity choices, opening the perspective of the potential competitiveness for realistic models.

Refined Schemes for Computing the Dynamics of Elastoviscoplastic Media

For a stable numerical solution of the system of equations governing an elastoviscoplastic continuous medium model, a second-order explicit-implicit scheme is proposed. An explicit approximation is used for the equations of motion, and an implicit approximation, for the constitutive relations containing a small relaxation time parameter in the denominator of the nonlinear free terms. A second-order implicit approximation for isotropic and anisotropic elastoviscoplastic models is constructed to match the orders of approximation of the explicit elastic and implicit adjustment steps. Refined formulas for correcting the stress deviators after the elastic step are derived for various viscosity function representations. The resulting solutions of the second-order implicit approximation for the stress deviators of the elastoviscoplastic equations admit passage to the limit as the relaxation time tends to zero. The correcting formulas obtained via this passage to the limit can be treated as regularizers of the numerical solutions to the elastoplastic systems.