Implicit-explicit Scheme Research Articles

Boundary effects on wave trains in the Exner model of sedimental transport

In this work we compute the numerical solution of the Exner model of sedimentation when a train of waves is imposed at the inflow boundary (E. Macca in Shock-Capturing methods: Well-Balanced Approximate Taylor and Semi-Implicit schemes. PhD thesis, Universit‘a degli Studi di Palermo, Palermo, (2022) and Major Revision, (2023)). The numerical solver is a second order finite-volume scheme, with semi-implicit time discretization based on Implicit-Explicit (IMEX) schemes, which guarantees better stability properties than explicit ones, still at a lower cost than fully implicit schemes. We show the effect of spurious reflected waves generated at the outflow edge of the computational domain, propose two remedies, and show how such spurious effects can be reduced by suitable non-reflecting boundary conditions.

A positivity-preserving implicit-explicit scheme with high order polynomial basis for compressible Navier–Stokes equations

In this paper, we are interested in constructing a scheme solving compressible Navier–Stokes equations, with desired properties including high order spatial accuracy, conservation, and positivity-preserving of density and internal energy under a standard hyperbolic type CFL constraint on the time step size, e.g., Δt=O(Δx). Strang splitting is used to approximate convection and diffusion operators separately. For the convection part, i.e., the compressible Euler equation, the high order accurate postivity-preserving Runge–Kutta discontinuous Galerkin method can be used. For the diffusion part, the equation of internal energy instead of the total energy is considered, and a first order semi-implicit time discretization is used for the ease of achieving positivity. A suitable interior penalty discontinuous Galerkin method for the stress tensor can ensure the conservation of momentum and total energy for any high order polynomial basis. In particular, positivity can be proven with Δt=O(Δx) if the Laplacian operator of internal energy is approximated by the Qk spectral element method with k=1,2,3. So the full scheme with Qk (k=1,2,3) basis is conservative and positivity-preserving with Δt=O(Δx), which is robust for demanding problems such as solutions with low density and low pressure induced by high-speed shock diffraction. Even though the full scheme is only first order accurate in time, numerical tests indicate that higher order polynomial basis produces much better numerical solutions, e.g., better resolution for capturing the roll-ups during shock reflection.

Uncoupling Techniques for Multispecies Diffusion–Reaction Model

We consider the multispecies model described by a coupled system of diffusion–reaction equations, where the coupling and nonlinearity are given in the reaction part. We construct a semi-discrete form using a finite volume approximation by space. The fully implicit scheme is used for approximation by time, which leads to solving the coupled nonlinear system of equations at each time step. This paper presents two uncoupling techniques based on the explicit–implicit scheme and the operator-splitting method. In the explicit–implicit scheme, we take the concentration of one species in coupling term from the previous time layer to obtain a linear uncoupled system of equations. The second approach is based on the operator-splitting technique, where we first solve uncoupled equations with the diffusion operator and then solve the equations with the local reaction operator. The stability estimates are derived for both proposed uncoupling schemes. We present a numerical investigation for the uncoupling techniques with varying time step sizes and different scales of the diffusion coefficient.

Unconditional stability of first and second orders implicit/explicit schemes for the natural convection equations

In this paper, we consider and analyze the stability of three implicit/explicit (IMEX) schemes for the time-dependent natural convection problem, these considered numerical schemes contain the first order backward Euler scheme, second order Crank-Nicolson IMEX scheme and BDF2-AB2 combination. All numerical schemes deal with the linear terms implicitly and the nonlinear terms explicitly. Then the original nonlinear problem is split into two linearized subproblems with constant coefficient matrixes, which reduces the computational complexity and saves a lot of storage. The biggest feature of this paper is to establish the unconditional stability of three IMEX schemes for incompressible flows, which improves and supplements the published theoretical findings. Finally, some numerical examples are provided to show the performances of the considered numerical schemes.

Explicit–Implicit Schemes for Calculating the Dynamics of Elastoviscoplastic Media with a Short Relaxation Time

Explicit–Implicit Schemes for Calculating the Dynamics of Elastoviscoplastic Media with a Short Relaxation Time

L ∞-norm convergence rates of an IMEX scheme for solving a partial integro-differential equation system arising from regime-switching jump-diffusion Asian option pricing

The purpose of this paper is to rigorously prove -norm convergence rates of an implicit-explicit (IMEX) difference method called Crank-Nicolson-Leap-Frog (CN-LF) scheme for solving a partial integro-differential equation (PIDE) system with moving boundaries from the regime-switching jump-diffusion Asian option pricing. The IMEX scheme is employed to discretize the PIDE system. Then the unconditional stability, unique solvability and convergence of second-order rates in both time and space are rigorously proved in the sense of -norm. Finally, several numerical examples are conducted to verify the theory.

Extended Lagrangian approach for the numerical study of multidimensional dispersive waves: Applications to the Serre-Green-Naghdi equations

In this paper we study two multidimensional nonlinear dispersive systems: the Serre-Green-Naghdi (SGN) equations describing dispersive shallow water flows, and the Iordanskii-Kogarko-Wijngaarden (IKW) equations describing fluids containing small compressible gas bubbles. These models are Euler-Lagrange equations for a given Lagrangian and share common mathematical structure, namely the dependence of the pressure on material derivatives of macroscopic variables. We develop a generic dispersive model such that SGN and IKW systems become its special cases if only one specifies the appropriate Lagrangian, and then use the extended Lagragian approach proposed in Favrie and Gavrilyuk (2017) to build its hyperbolic approximation. The new approximate model is unconditionally hyperbolic for both SGN and IKW cases, and accurately describes dispersive phenomena, which allows to impose discontinuous initial data and study dispersive shock waves. We consider the 2-D hyperbolic version of SGN system as an example for numerical simulations and apply a second order implicit-explicit scheme in order to numerically integrate the system. The obtained 1-D and 2-D results are in close agreement with available exact solutions and numerical tests.

Fractured inclusion localization and characterization based on deep convolutional neural networks

Seismic exploration is a standard method of prospecting for hydrocarbon deposits. The major goal is to correctly localize the spatial position of the fractured inclusion and estimate its parameters. In this work we investigate the applicability of machine-learning algorithms for this problem. To generate synthetic seismograms the model of a deformable solid medium containing slip planes with nonlinear slip conditions on them is used. The explicit-implicit scheme is applied for obtaining the numerical solution of a constitutive system of equations. The problem of the seismic wave propagation in an inhomogeneous fractured geological model based on the well-known Marmousi2 model in a two-dimensional case is considered. Deep convolutional neural networks are used to provide a fast solution for the inverse problem of restoring the parameters of the fractured inclusion based on the surface measurements. The neural network architecture follows a segmentation approach and targets primarily a spatial localization of the fractured inclusion. However, the mechanical parameters of the inclusion are also estimated using a single run of the same network.

An acoustic-transport splitting method for the barotropic Baer-Nunziato two-phase flow model

This work focuses on the numerical approximation of the barotropic Baer-Nunziato two-phase flow model. We propose a numerical scheme that relies on an operator splitting method corresponding to a separate treatment of the acoustic and the material transport phenomena. In the subsonic case, this also corresponds to a separate treatment of the fast and the slow propagation phenomena. This approach follows the lines of the implicit-explicit schemes developed in [8]. The operator splitting enable the use of time steps that are no longer constrained by the sound velocity thanks to an implicit treatment of the acoustic waves, while maintaining accuracy in the subsonic regime thanks to an explicit treatment of the material waves. In the present setting, a particular attention will be also given to the discretization of the non-conservative terms that figure in the two-phase model. We prove that the proposed numerical strategy is positivity preserving for the volume fractions and the partial masses. The scheme is tested against several one-dimensional test cases including flows featuring vanishing phases.

Adjoint-based optimization of two-dimensional Stefan problems

A range of optimization cases of two-dimensional Stefan problems, solved using a tracking-type cost-functional, is presented. A level-set method is used to capture the interface between the liquid and solid phases and an immersed boundary (cut cell) method coupled with an implicit time-advancement scheme is employed to solve the heat equation. A conservative implicit-explicit scheme is then used for solving the level-set transport equation. The resulting numerical framework is validated with respect to existing analytical solutions of the forward Stefan problem. An adjoint-based algorithm is then employed to efficiently compute the gradient used in the optimization algorithm (L-BFGS). The algorithm follows a continuous adjoint framework, where adjoint equations are formally derived using shape calculus and transport theorems. A wide range of control objectives are presented, and the results show that using parameterized boundary actuation leads to effective control strategies in order to suppress interfacial instabilities or to maintain a desired crystal shape.

Partial learning using partially explicit discretization for multicontinuum/multiscale problems. Fractured poroelastic media simulation

The poroelasticity problem plays a vital role in various fields of science and technology. For example, it has applications in the petroleum industry, agricultural science, and biomedicine. The mathematical model consists of a coupled system of differential equations for mechanical displacements and pressure.In this paper, we propose a new approach for solving the poroelasticity problem in a fractured medium based on hybrid explicit–implicit learning (HEI). For spatial approximation, we use the finite element method with standard linear basis functions. We consider a poroelastic medium with large hydraulic fractures and use the Discrete Fracture Model. For the time approximation, we apply the explicit–implicit scheme. The explicit scheme is used for the porous matrix, which is a low-conductive continuum. While for fractures, we apply the implicit scheme due to its high conductivity. To facilitate the calculations, we use the fixed strain and fixed stress splitting schemes. The main idea of the proposed method lies in partial learning. Full training of the numerical solution is extremely difficult due to a large number of degrees of freedom, so we propose to train a part of the solution. Large hydraulic fractures have few degrees of freedom, but they are highly conductive and require a more costly implicit scheme. Therefore, we train the neural network to generate pressure in the fractures at several time points and then interpolate for other times. Thus, we treat the fracture pressure as a known function and use it to find the pressure in the porous matrix. Numerical results for a two-dimensional model problem are presented.

Numerical schemes for a multi-species BGK model with velocity-dependent collision frequency

We consider a kinetic description of multi-species gas mixture modeled with Bhatnagar-Gross-Krook (BGK) collision operators, in which the collision frequency varies not only in time and space but also with the microscopic velocity. In this model, the Maxwellians typically used in standard BGK operators are replaced by a generalization of such target functions, which are defined by a variational procedure [20]. In this paper we present a numerical method for simulating this model, which uses an Implicit-Explicit (IMEX) scheme to minimize a certain potential function, mimicking the Lagrange functional that appears in the theoretical derivation. We show that theoretical properties such as conservation of mass, total momentum and total energy as well as positivity of the distribution functions are preserved by the numerical method, and illustrate its usefulness and effectiveness with numerical examples.

A one-dimensional field dislocation mechanics model using discontinuous Galerkin method

A numerical solution strategy for a one-dimensional field dislocation mechanics (FDM) model using the Discontinuous Galerkin (DG) method is developed. The FDM model is capable of simulating the dynamics of discrete, nonsingular dislocations using a partial differential equation involving a conservation law for the Burgers vector content with constitutive input for nucleation and velocity. Modeling of individual dislocation lines with an equilibrium compact core structure in the context of this continuum elastoplastic framework requires a non-convex stored energy density. Permanent deformation and stress redistribution caused by the dissipative transport of dislocations is modeled using thermodynamics-based constitutive laws. A DG method is employed to discretize the evolution equation of dislocation density yielding high orders of accuracy when the solution is smooth. The trade-offs of using a high order explicit Runge–Kutta time stepping and an implicit–explicit scheme are discussed. The developed numerical scheme is used to simulate the transport of a single screw dislocation wall in the case of a non-zero applied strain.

Odd-Even based asymptotic preserving scheme for a 2D stochastic kinetic–fluid model

A numerical approach to solve a new coupled stochastic kinetic–fluid model is proposed in this paper. The approach yields a stable numerical scheme in the hydrodynamic limit as the Knudsen number goes to zero along the transition from kinetic regime to macroscopic regime. The method is based on the Odd-Even decomposition of the kinetic stochastic chemotaxis–fluid model which leads to an implicit-explicit scheme based on a two steps splitting strategy. The method is numerically validated by various tests in two dimensional space.

Implicit-explicit second derivative general linear methods with strong stability preserving explicit part

In this paper, we discuss the class of implicit–explicit (IMEX) methods for systems of ordinary differential equations where the explicit part has strong stability preserving (SSP) property, while the implicit part of the method has Runge–Kutta stability property together with A-, or L-stability. The explicit part of these methods is treated by the explicit second derivative general linear methods and the implicit part is treated by the implicit general linear methods. We construct methods of order p=s and stage order q=p and q=p−1 with r=2, of order p=q=s−1=r−1 and of order p=q+1=s−1=r−1 up to the order p=4 with large SSP coefficients with respect to the large region of absolute stability when the implicit and explicit parts interact with each other. In the case of methods with p=q=r−1=s−1 and p=q+1=s−1=r−1, the implicit part has inherent Runge–Kutta stability property, and is L-stable. Finally, we test the proposed SSP IMEX schemes on some one dimensional linear and nonlinear problems, and the results for the expected order of convergence are presented.

A generalized SAV approach with relaxation for dissipative systems

The scalar auxiliary variable (SAV) approach [31] and its generalized version GSAV proposed in [20] are very popular methods to construct efficient and accurate energy stable schemes for nonlinear dissipative systems. However, the discrete value of the SAV is not directly linked to the free energy of the dissipative system, and may lead to inaccurate solutions if the time step is not sufficiently small. Inspired by the relaxed SAV method proposed in [21] for gradient flows, we propose in this paper a generalized SAV approach with relaxation (R-GSAV) for general dissipative systems. The R-GSAV approach preserves all the advantages of the GSAV approach, in addition, it dissipates a modified energy that is directly linked to the original free energy. We prove that the k-th order implicit-explicit (IMEX) schemes based on R-GSAV are unconditionally energy stable, and we carry out a rigorous error analysis for k=1,2,3,4,5. We present ample numerical results to demonstrate the improved accuracy and effectiveness of the R-GSAV approach.

Stability and Error Analysis of IMEX SAV Schemes for the Magneto-Hydrodynamic Equations

We construct and analyze first- and second-order implicit-explicit schemes based on the scalar auxiliary variable approach for the magneto-hydrodynamic equations. These schemes are linear, only require solving a sequence of linear differential equations with constant coefficients at each time step, and are unconditionally energy stable. We derive rigorous error estimates for the velocity, pressure, and magnetic field of the first-order scheme in the two-dimensional case without any condition on the time step. Numerical examples are presented to validate the proposed schemes.

A New Parallelized Computation Method of HASC-N Difference Scheme for Inhomogeneous Time Fractional Fisher Equation

The fractional Fisher equation has a wide range of applications in many engineering fields. The rapid numerical methods for fractional Fisher equation have momentous scientific meaning and engineering applied value. A parallelized computation method for inhomogeneous time-fractional Fisher equation (TFFE) is proposed. The main idea is to construct the hybrid alternating segment Crank-Nicolson (HASC-N) difference scheme based on alternating segment difference technology, using the classical explicit scheme and classical implicit scheme combined with Crank-Nicolson (C-N) scheme. The unique existence, unconditional stability and convergence are proved theoretically. Numerical tests show that the HASC-N difference scheme is unconditionally stable. The HASC-N difference scheme converges to O(τ2−α+h2) under strong regularity and O(τα+h2) under weak regularity of fractional derivative discontinuity. The HASC-N difference scheme has high precision and distinct parallel computing characteristics, which is efficient for solving inhomogeneous TFFE.

Energy stable L2 schemes for time-fractional phase-field equations

In this work, we establish the energy stability of high-order L2-type schemes for time-fractional phase-field equations. We propose a reformulation of the discrete L2 operator, show the monotonicity of some associated coefficients, and then obtain two positive definiteness properties for the L2 operator. Based on these two properties, we show that the energy is bounded by the initial energy for the L2 scalar auxiliary variable schemes of time-fractional gradient flows. Furthermore, a fractional energy law can be established for the L2 implicit-explicit scheme of the time-fractional Allen–Cahn equation. Several numerical experiments are provided to verify the stability as well as the convergence.

Effects of landing gear layout on the safe rollout envelope of equipped–skid aircraft

Rollout is an inevitable phase of realizing horizontal landing for an equipped–skid aircraft; therefore, one should pay attention to the safety of the entire rollout process. The research presented herein analyzes the effects of the landing gear layout on the safe rollout envelope, which is the set of all states where the aircraft can remain in the desired state region during the entire rollout phase. First, the nonlinear reachability analysis and its internal relation with a safe envelope are briefly described. To improve the calculation efficiency of approximating the safe envelope, a mixed implicit explicit (MIE) scheme is introduced. Subsequently, a simple yet accurate model of nonlinear aircraft dynamics is established and the dangerous state region during the rollout phase is defined. Finally, based on the numerical results, the effects of the layout type on the safe envelope are discussed. In addition, a reasonable safety evaluation index is proposed to quantitatively analyze the effects of position parameters on the rollout safety. The results indicate that with the application of the variable friction skid, the safe envelope of the wheel–skid aircraft is larger than that of the all–skid aircraft. Compared with the original position parameters, the optimized values increase the rollout safety of the wheel–skid aircraft by 4.18%.