Implicit-explicit Research Articles

Nonintrusive model order reduction for cross-diffusion systems

In this paper, we investigate tensor based nonintrusive reduced-order models (ROMs) for parametric cross-diffusion equations. The full-order model (FOM) consists of ordinary differential equations (ODEs) in matrix or tensor form resulting from finite difference discretization of the differential operators by taking the advantage of Kronecker structure. The matrix/tensor differential equations are integrated in time with the implicit–explicit (IMEX) Euler method. The reduced bases, relying on a finite sample set of parameter values, are constructed in form of a two-level approach by applying higher-order singular value decomposition (HOSVD) to the space–time snapshots in tensor form, which leads to a large amount of computational and memory savings. The nonintrusive reduced approximation for an arbitrary parameter value is obtained through tensor product of the reduced basis by the parameter dependent core tensor that contains the reduced coefficients. The reduced coefficients for new parameter values are computed with the radial basis functions. The efficiency of the proposed method is illustrated through numerical experiments for two-dimensional Schnakenberg and three-dimensional Brusselator cross-diffusion equations. The spatiotemporal patterns are accurately predicted by the reduced-order models with speedup factors of orders two and three over the full-order models.

TVD-MOOD schemes based on implicit-explicit time integration

The context of this work is the development of first order total variation diminishing (TVD) implicit-explicit (IMEX) Runge-Kutta (RK) schemes as a basis of a Multidimensional Optimal Order detection (MOOD) approach to approximate the solution of hyperbolic multi-scale equations. A key feature of our newly proposed TVD schemes is that the resulting CFL condition does not depend on the fast waves of the considered model, as long as they are integrated implicitly. However, a result from Gottlieb et al. [1] gives a first order barrier for unconditionally stable implicit TVD-RK schemes and TVD-IMEX-RK schemes with scale-independent CFL conditions. Therefore, the goal of this work is to consistently improve the resolution of a first-order IMEX-RK scheme, while retaining its L∞ stability and TVD properties. In this work we present a novel approach based on a convex combination between a first-order TVD IMEX Euler scheme and a potentially oscillatory high-order IMEX-RK scheme. We derive and analyse the TVD property for a scalar multi-scale equation and numerically assess the performance of our TVD schemes compared to standard L-stable and SSP IMEX RK schemes from the literature. Finally, the resulting TVD-MOOD schemes are applied to the isentropic Euler equations.

High-order discontinuous Galerkin approximation for a three-phase incompressible Navier–Stokes/Cahn–Hilliard model

In this work we introduce the development of a three-phase incompressible Navier–Stokes/Cahn–Hilliard numerical method to simulate three-phase flows, present in many industrial operations. The numerical method is then applied to successfully solve oil transport problems, such as those found in the oil and gas industry. The three-phase model adopted in this work is a Cahn–Hilliard diffuse interface model, which was derived by Boyer and Lapuerta (2006). The Cahn–Hilliard model is coupled to the kinetic-energy stable incompressible Navier–Stokes equations model derived by Manzanero et al. (2020). The spatial discretization uses a high-order discontinuous Galerkin spectral element method which yields highly accurate results in arbitrary geometries. An implicit–explicit (IMEX) method is adopted as temporal scheme for the Cahn–Hilliard equation, while Runge–Kutta 3 (RK3) is used for the Navier–Stokes equations. The developed numerical tool is validated with a manufactured solution test case and used to simulate multiphase flows in pipes, including and a three-phase T-shaped pipe intersection.

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.

Error estimates for the optimal control of Navier-Stokes equations using curvature based stabilization

This paper presents a family of implicit-explicit (IMEX) time stepping scheme for the optimal control problem of the unsteady Navier-Stokes equations (NSE). The main feature of this kind of optimal control problem is that stabilization terms are proportional to discrete curvature of the solutions. First and second order optimality conditions are used for optimality of the curvature based stabilized Navier-Stokes equations. Complete stability and error analyses of state, adjoint and control variables are presented. Numerical experiments verify theoretical findings and illustrate the improvement of approximate solutions enhancing the efficiency of numerical scheme.

Stability of high order finite difference and local discontinuous Galerkin schemes with explicit-implicit-null time-marching for high order dissipative and dispersive equations

Time discretization is an important issue for time-dependent partial differential equations (PDEs). For the k-th (k≥2) order PDEs, the explicit time-marching method may suffer from a severe time step restriction τ=O(hk) for stability. The implicit and implicit-explicit (IMEX) time-marching methods can overcome this constraint. However, for the equations with nonlinear high derivative terms, the IMEX methods are not good choices either, since a nonlinear algebraic system must be solved (e.g. by Newton iteration) at each time step. The explicit-implicit-null (EIN) time-marching method is designed to cope with the above mentioned shortcomings. The basic idea of the EIN method discussed in this paper is to add and subtract a sufficiently large linear highest derivative term on one side of the considered equation, and then apply the IMEX time-marching method to the equivalent equation. The EIN method so designed does not need any nonlinear iterative solver, and the severe time step restriction for explicit methods can be removed. Coupled with the EIN time-marching method, we will discuss the high order finite difference and local discontinuous Galerkin schemes for solving high order dissipative and dispersive equations, respectively. By the aid of the Fourier method, we perform stability analysis for the schemes on the simplified equations with periodic boundary conditions, which demonstrates the stability criteria for the resulting schemes. Even though the analysis is only performed on the simplified equations, numerical experiments show that the proposed schemes are stable and can achieve optimal orders of accuracy for both one-dimensional and two-dimensional linear and nonlinear equations.

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.

High order well-balanced asymptotic preserving finite difference WENO schemes for the shallow water equations in all Froude numbers

In this paper, high order semi-implicit well-balanced and asymptotic preserving finite difference WENO schemes are proposed for the shallow water equations with a non-flat bottom topography. We consider the Froude number ranging from O(1) to 0, which in the zero Froude limit becomes the “lake equations” for balanced flow without gravity waves. We apply a well-balanced finite difference WENO reconstruction, coupled with a stiffly accurate implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting semi-implicit scheme can be shown to be well-balanced, asymptotic preserving (AP) and asymptotically accurate (AA) at the same time. Both one- and two-dimensional numerical results are provided to demonstrate the high order accuracy, AP property and good performance of the proposed methods in capturing small perturbations of steady state solutions.

High‐order implicit‐explicit additive Runge–Kutta schemes for numerical combustion with adaptive mesh refinement

AbstractThe objective of this study is to develop and apply efficient solution techniques for numerical modeling of combustion with stiff chemical kinetics in practical combustors. The new technique combines a fourth‐order implicit‐explicit (IMEX) additive Runge‐Kutta scheme (ARK) with adaptive mesh refinement (AMR). The IMEX component treats the stiff reactions implicitly but integrates convection and diffusion explicitly in time, and thus permits the solution to advance with larger time‐step sizes than that of explicit time‐marching methods alone. The AMR further adds computational efficiency by effectively placing high spatial resolution meshes in regions with strong gradients, such as flame fronts. The novelty of this study is in the integration of a fourth‐order IMEX ARK method with AMR for a high‐order finite‐volume scheme and the application to solving complex reacting flows governed by the compressible Navier–Stokes equations with very stiff chemistry in a practical combustor geometry. The effectiveness and performance of the adaptive ARK4 is assessed for complex reacting flows by examining properties, such as the presence of shock waves, the time‐scale changes in response to AMR levels, and the size and stiffness of reaction mechanisms for various fuels such as , , and . The new adaptive ARK4 method is verified and validated using a convection‐diffusion‐reaction problem and shock‐driven combustion, respectively. The validated algorithm is then applied to solve the stiff ‐air combustion in a bluff‐body combustor. A significant speedup of three orders of magnitude is achieved in comparison to the standard ERK4 method at the given solution accuracy.

Conservative DG method for the micro-macro decomposition of the Vlasov–Poisson–Lenard–Bernstein model

The micro-macro (mM) decomposition approach is considered for the numerical solution of the Vlasov–Poisson–Lenard–Bernstein (VPLB) system, which is relevant for plasma physics applications. In the mM approach, the kinetic distribution function is decomposed as f=E[ρf]+g, where E is a local equilibrium distribution, depending on the macroscopic moments ρf=∫Refdv=〈ef〉R, where e=(1,v,12v2)T, and g, the microscopic distribution, is defined such that 〈eg〉R=0. We aim to design numerical methods for the mM decomposition of the VPLB system, which consists of coupled equations for ρf and g. To this end, we use the discontinuous Galerkin (DG) method for phase-space discretization, and implicit-explicit (IMEX) time integration, where the phase-space advection terms are integrated explicitly and the collision operator is integrated implicitly. We give special consideration to ensure that the resulting mM method maintains the 〈eg〉R=0 constraint, which may be necessary for obtaining (i) satisfactory results in the collision dominated regime with coarse velocity resolution, and (ii) unambiguous conservation properties. The constraint-preserving property is achieved through a consistent discretization of the equations governing the micro and macro components. We present numerical results that demonstrate the performance of the mM method. The mM method is also compared against a corresponding DG-IMEX method solving directly for f.

A p‐adaptive, implicit‐explicit mixed finite element method for diffusion‐reaction problems

AbstractA new class of implicit‐explicit (IMEX) methods combined with a ‐adaptive mixed finite element formulation is proposed to simulate the diffusion of reacting species. Hierarchical polynomial functions are used to construct a conforming base for the flux vectors, and a non‐conforming base for the mass concentration of the species. The mixed formulation captures the distinct nonlinearities associated with the flux constitutive equations and the reaction terms. The IMEX method conveniently treats these two sources of nonlinearity implicitly and explicitly, respectively, within a single time‐stepping framework. A reliable a posteriori error estimate is proposed and analyzed. A ‐adaptive algorithm based on the proposed a posteriori error estimate is also constructed. The combination of the proposed residual‐based a posteriori error estimate and hierarchical finite element spaces allows for the formulation of an efficient ‐adaptive algorithm. A series of numerical examples demonstrate the performance of the approach for problems involving travelling waves, and possessing discontinuities and singularities. The flexibility of the formulation is illustrated via selected applications in pattern formation and electrophysiology.

A cardiac electromechanical model coupled with a lumped-parameter model for closed-loop blood circulation

We propose a novel mathematical and numerical model for cardiac electromechanics, wherein biophysically detailed core models describe the different physical processes concurring to the cardiac function. The core models, once suitably approximated, are coupled by a computationally efficient strategy. Our model is based on: (1) the combination of implicit-explicit (IMEX) schemes to solve the different core cardiac models, (2) an Artificial Neural Network based model, that surrogates a biophysically detailed but computationally demanding microscale model of active force generation and (3) appropriate partitioned schemes to couple the different models in this multiphysics setting. We employ a flexible and scalable intergrid transfer operator, which allows to interpolate Finite Element functions between nested meshes and, possibly, among arbitrary Finite Element spaces for the different core models. Our core 3D electromechanical model of the left ventricle is coupled with a closed-loop 0D model of the vascular network (and the other cardiac chambers) by an approach that is energy preserving. More precisely, we derive a balance law for the mechanical energy of the whole circulatory network. This provides a quantitative insight into the energy utilization, dissipation and transfer among the different compartments of the cardiovascular network and during different stages of the heartbeat. On this ground, a new tool is proposed to validate some energy indicators adopted in the daily clinical practice. A further contribution of this paper is the proposition of a robust algorithm for the reconstruction of the stress-free reference configuration. This feature is fundamental to correctly initialize our electromechanical simulations. As a matter of fact, the geometry acquired from medical imaging typically refers to a configuration affected by residual internal stresses, whereas the elastodynamics equations that govern the mechanics core model are related to a stress-free configuration. To prove the biophysical accuracy of our computational model, we address different scenarios of clinical interest, namely by varying preload, afterload and contractility.

A new class of implicit–explicit BDF[formula omitted] SAV schemes for general dissipative systems and their error analysis

We construct a new class of efficient implicit–explicit (IMEX) BDFk schemes combined with a scalar auxiliary variable (SAV) approach for general dissipative systems. We show that these schemes are unconditionally stable, and lead to a uniform bound of the numerical solution in the norm based on the principal linear operator in the free energy. Based on this uniform bound, we carry out a rigorous error analysis for the kth-order (k=1,2,3,4,5) SAV schemes in a unified form for a class of general dissipative systems. We also present numerical results confirming our theoretical convergence rates.

On the Construction of Conservative Semi-Lagrangian IMEX Advection Schemes for Multiscale Time Dependent PDEs

This article is devoted to the construction of a new class of semi-Lagrangian (SL) schemes with implicit-explicit (IMEX) Runge-Kutta (RK) time stepping for PDEs involving multiple space-time scales. The semi-Lagrangian (SL) approach fully couples the space and time discretization, thus making the use of RK strategies particularly difficult to be combined with. First, a simple scalar advection-diffusion equation is considered as a prototype PDE for the development of a high order formulation of the semi-Lagrangian IMEX algorithms. The advection part of the PDE is discretized explicitly at the aid of a SL technique, while an implicit discretization is employed for the diffusion terms. In this way, an unconditionally stable numerical scheme is obtained, that does not suffer any CFL-type stability restriction on the maximum admissible time step. Second, the SL-IMEX approach is extended to deal with hyperbolic systems with multiple scales, including balance laws, that involve shock waves and other discontinuities. A conservative scheme is then crucial to properly capture the wave propagation speed and thus to locate the discontinuity and the plateau exhibited by the solution. A novel SL technique is proposed, which is based on the integration of the governing equations over the space-time control volume which arises from the motion of each grid point. High order of accuracy is ensured by the usage of IMEX RK schemes combined with a Cauchy–Kowalevskaya procedure that provides a predictor solution within each space-time element. The one-dimensional shallow water equations (SWE) are chosen to validate the new conservative SL-IMEX schemes, where convection and pressure fluxes are treated explicitly and implicitly, respectively. The asymptotic-preserving (AP) property of the novel schemes is also studied considering a relaxation PDE system for the SWE. A large suite of convergence studies for both the non-conservative and the conservative version of the novel class of methods demonstrates that the formal order of accuracy is achieved and numerical evidences about the conservation property are shown. The AP property for the corresponding relaxation system is also investigated.

High Order Strong Stability Preserving MultiDerivative Implicit and IMEX Runge--Kutta Methods with Asymptotic Preserving Properties

In this work we present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge--Kutta schemes and SSP implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order $p>2$ is unique among SSP methods and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge--Kutta schemes of order up to $p=4$ and IMEX Runge--Kutta schemes of order up to $p=3$. For the multiderivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar--Gross--Krook kinetic equation. We present numerical results to support the theoretical results on a variety of problems.

Numerical approximation to semi-linear stiff neutral equations via implicit–explicit general linear methods

Among the initial value problems of semi-linear neutral equations, there are a class of so-called stiff problems, where the classical explicit methods do not work as the methods have only bounded stability regions, which confine the computational stepsize to be excessively small and thus leads to an unsuccessful calculation. For resolving this difficult issue, ones turned to develop the implicit methods with unbounded stability regions to solve the stiff problems. Nevertheless, it is well-known that the implementation of an implicit method needs a large computational cost. In order to improve the computational efficiency, in Refs. Tan and Zhang (2018, 2020), the authors adopted the implicit–explicit (IMEX) splitting technique to derive the extended IMEX one-leg methods and IMEX Runge–Kutta methods, respectively. Unfortunately, these two methods have the serious order barrier. So far, for stiff neutral equations (SNEs), no IMEX method with order more than two has been found. To improve the computational accuracy and efficiency of IMEX methods, in the present paper, we construct a class of extended implicit–explicit general linear (EIEGL) methods for solving semi-linear SNEs. Under some suitable conditions, an EIEGL method is proved to be stable and convergent of order p whenever the underlying implicit–explicit general linear (IEGL) method has order p and stage order p. With applications to several concrete problems of SNEs, the computational accuracy of EIEGL methods are further illustrated, where we also verify that the convergence order of EIEGL methods can exceed two, namely, third- and fourth-order EIEGL methods can be obtained. Moreover, based on a numerical comparison with the extended implicit general linear (EIGL) methods, the advantage of EIEGL methods in computational efficiency is shown.

An Eulerian-Lagrangian Runge-Kutta Finite Volume (El-Rk-Fv) Method for Solving Convection and Convection-Diffusion Equations

We propose a new Eulerian-Lagrangian Runge-Kutta finite volume method for numerically solving convection and convection-diffusion equations. Eulerian-Lagrangian and semi-Lagrangian methods have grown in popularity mostly due to their ability to allow large time steps. Our proposed scheme is formulated by integrating the PDE on a space-time region partitioned by approximations of the characteristics determined from the Rankine-Hugoniot jump condition; and then rewriting the time-integral form into a time differential form to allow application of Runge-Kutta (RK) methods via the method-of-lines approach. The scheme can be viewed as a generalization of the standard Runge-Kutta finite volume (RK-FV) scheme for which the space-time region is partitioned by approximate characteristics with zero velocity. The high-order spatial reconstruction is achieved using the recently developed weighted essentially non-oscillatory schemes with adaptive order (WENO-AO); and the high-order temporal accuracy is achieved by explicit RK methods for convection equations and implicit-explicit (IMEX) RK methods for convection-diffusion equations. Our algorithm extends to higher dimensions via dimensional splitting. Numerical experiments demonstrate our algorithm's robustness, high-order accuracy, and ability to handle extra large time steps.

A Generalized Sav Approach with Relaxation for Dissipative Systems

The scalar auxiliary variable (SAV) approach \cite{shen2018scalar} and its generalized version GSAV proposed in \cite{huang2020highly} 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 \cite{jiang2022improving} 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 appraoch, 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.

Nonintrusive Model Order Reduction for Cross-Diffusion Systems

In this paper, we investigate tensor based nonintrusive reduced-order models (ROMs) for parametric cross-diffusion equations. The full-order model (FOM) consists of ordinary differential equations (ODEs) in matrix or tensor form resulting from finite-difference discretization of the differential operators by taking the advantage of Kronecker structure. The matrix/tensor differential equations are integrated in time with the implicit-explicit (IMEX) Euler method. The reduced bases, relying on a finite sample set of parameter values, are constructed in form of a two-level approach by applying higher-order singular value decomposition (HOSVD) to the space-time snapshots in tensor form, which leads to a large amount of computational and memory savings. The nonintrusive reduced approximations for an arbitrary parameter value are obtained through tensor product of the reduced basis by the parameter dependent core tensor that contains the reduced coefficients. The reduced coefficients for new parameter values are computed using radial basis function (RBF) interpolation. The efficiency of the proposed method is illustrated through numerical experiments for two-dimensional Schnakenberg and three-dimensional Brusselator cross-diffusion equations. The spatiotemporal patterns are accurately predicted by the reduced-order models with speed-up factors of orders two and three over the full-order models.

A unified asymptotic preserving and well-balanced scheme for the Euler system with multiscale relaxation

The design and analysis of a unified asymptotic preserving (AP) and well-balanced scheme for the Euler Equations with gravitational and frictional source terms is presented in this paper. The asymptotic behaviour of the Euler system in the limit of zero Mach and Froude numbers, and large friction is characterised by an additional scaling parameter. Depending on the values of this parameter, the Euler system relaxes towards a hyperbolic or a parabolic limit equation. Standard Implicit–Explicit Runge–Kutta schemes are incapable of switching between these asymptotic regimes. We propose a time semi-discretisation to obtain a unified scheme which is AP for the two different limits. A further reformulation of the semi-implicit scheme can be recast as a fully-explicit method in which the mass update contains both hyperbolic and parabolic fluxes. A space–time fully-discrete scheme is derived using a finite volume framework. A hydrostatic reconstruction strategy, an upwinding of the sources at the interfaces, and a careful choice of the central discretisation of the parabolic fluxes are used to achieve the well-balancing property for hydrostatic steady states. Results of several numerical case studies are presented to substantiate the theoretical claims and to verify the robustness of the scheme.