Order Implicit-explicit Research Articles

A new mixed Boltzmann-BGK model for mixtures solved with an IMEX finite volume scheme on unstructured meshes

In this work, we consider a novel model for a binary mixture of inert gases. The model, which preserves the structure of the original Boltzmann equations, combines integro-differential collision operators with BGK relaxation terms in each kinetic equation: the first involving only collisions among particles of the same species, while the second ones taking into account the inter–species interactions. We prove consistency of the model: conservation properties, positivity of all temperatures, H-theorem and convergence to a global equilibrium in the shape of a global Maxwell distribution. We also derive hydrodynamic equations under different collisional regimes. In a second part, to numerically solve the governing equations, we introduce a class of time and space high order finite volume schemes that are able to capture the behaviors of the different hydrodynamic limit models: the classical Euler equations as well as the multi-velocities and temperatures Euler system. The methods work by integrating the distribution functions over arbitrarily shaped and closed control volumes in 2D using Central Weighted ENO (CWENO) techniques and make use of spectral methods for the approximation of the Boltzmann integrals with high order Implicit-Explicit (IMEX) Runge Kutta schemes. For these methods, we prove accuracy and preservation of the discrete asymptotic states. In the numerical section we first show that the methods indeed possess the theoretical order of accuracy for different regimes and second we analyse their capacity in solving different two dimensional problems arising in kinetic theory. To speed up the computational time, all simulations are run with MPI parallelization on 64 cores, thus showing the potentiality of the proposed methods to be used for HPC (High Performance Computing) on massively parallel architectures.

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.

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.

High order finite volume schemes with IMEX time stepping for the Boltzmann model on unstructured meshes

In this work, we present a family of time and space high order finite volume schemes for the solution of the full Boltzmann equation. The velocity space is approximated by using a discrete ordinate approach while the collisional integral is approximated by spectral methods. The space reconstruction is implemented by integrating the distribution function, which describes the state of the system, over arbitrarily shaped and closed control volumes using a Central Weighted ENO (CWENO) technique. Compared to other reconstruction methods, this approach permits to keep compact stencil sizes which is a remarkable property in the context of kinetic equations due to the considerable demand of computational resources. The full discretization is then obtained by combining the previous phase-space approximation with high order Implicit–Explicit (IMEX) Runge–Kutta schemes. These methods guarantee stability, accuracy and preservation of the asymptotic state. Comparisons of the Boltzmann model with simpler relaxation type kinetic models (like BGK) are proposed showing the capability of the Boltzmann equation to capture different physical solutions. The theoretical order of convergence is numerically measured in different regimes and the methods are tested on several standard two-dimensional benchmark problems in comparison with Direct Simulation Monte Carlo results. The article ends with a prototype engineering problem consisting of a subsonic and a supersonic flow around a NACA 0012 airfoil. All test cases are run with MPI parallelization on several cores, thus making the proposed methods suitable for parallel distributed memory supercomputers.

A discontinuous Galerkin approximation for a wall–bounded consistent three–component Cahn–Hilliard flow model

We present a high–order discontinuous Galerkin (DG) discretization for the three–phase Cahn–Hilliard model of [Boyer, F., & Lapuerta, C. (2006). Study of a three component Cahn–Hilliard flow model]. In this model, consistency is ensured with an additional term in the chemical free–energy. The model considered in this work includes a wall boundary condition that allows for an arbitrary equilibrium contact angle in three–phase flows. The model is discretized with a high–order discontinuous Galerkin spectral element method that uses the symmetric interior penalty to compute the interface fluxes, and allows for unstructured meshes with curvilinear hexahedral elements. The integration in time uses a first order IMplicit–EXplicit (IMEX) method, such that the associated linear systems are decoupled for the two Cahn–Hilliard equations. Additionally, the Jacobian matrix is constant, and identical for both equations. This allows us to solve the two systems by performing only one LU factorization, with the size of the two–phase system, followed by two Gauss substitutions. Finally, we test numerically the accuracy of the scheme providing convergence analyses for two and three–dimensional cases, including the captive bubble test, the study of two bubbles in contact with a wall and the spinodal decomposition in a cube and in a curved pipe with a “T” junction.

An embedded variable step IMEX scheme for the incompressible Navier–Stokes equations

This report presents a series of implicit–explicit (IMEX) variable stepsize algorithms for the incompressible Navier–Stokes equations (NSE). Here, IMEX means the nonlinear term is treated fully explicitly while the remaining terms are treated implicitly. With the advent of new computer architectures there has been growing demand for low memory solvers of this type. The addition of time adaptivity improves the accuracy and greatly enhances the efficiency of the algorithm. We prove energy stability of an embedded first–second order IMEX pair. For the first order member of the pair, we prove stability for variable stepsizes, and analyze convergence. We believe this to be the first proof of this type for a variable stepsize IMEX scheme for the incompressible NSE. We then define and test a variable stepsize, variable order IMEX scheme using these methods. Our work contributes several firsts for IMEX NSE schemes, including an energy argument and error analysis of a two-step, variable stepsize method, and embedded error estimation for an IMEX multi-step method.

High order semi-implicit weighted compact nonlinear scheme for the all-Mach isentropic Euler system

The computation of compressible flows at all Mach numbers is a very challenging problem. An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime, while it can deal with stiffness and accuracy in the low Mach number regime. This paper designs a high order semi-implicit weighted compact nonlinear scheme (WCNS) for the all-Mach isentropic Euler system of compressible gas dynamics. To avoid severe Courant-Friedrichs-Levy (CFL) restrictions for low Mach flows, the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components. A third-order implicit-explicit (IMEX) method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives. The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit. One- and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.

A free–energy stable nodal discontinuous Galerkin approximation with summation–by–parts property for the Cahn–Hilliard equation

We present a nodal Discontinuous Galerkin (DG) scheme for the Cahn–Hilliard equation that satisfies the summation–by–parts simultaneous–approximation–term (SBP–SAT) property. The latter permits us to show that the discrete free–energy is bounded, and as a result, the scheme is provably stable. The scheme and the stability proof are presented for general curvilinear three–dimensional hexahedral meshes. We use the Bassi–Rebay 1 (BR1) scheme to compute interface fluxes, and a first order IMplicit–EXplicit (IMEX) scheme to integrate in time. We provide a semi–discrete stability study, and a fully–discrete proof subject to a positivity condition on the solution. Lastly, we test the theoretical findings using numerical cases that include two and three–dimensional problems.

Unconditional Energy Stability Analysis of a Second Order Implicit–Explicit Local Discontinuous Galerkin Method for the Cahn–Hilliard Equation

In this article, we present a second-order in time implicit–explicit (IMEX) local discontinuous Galerkin (LDG) method for computing the Cahn–Hilliard equation, which describes the phase separation phenomenon. It is well-known that the Cahn–Hilliard equation has a nonlinear stability property, i.e., the free-energy functional decreases with respect to time. The discretized Cahn–Hilliard system modeled by the IMEX LDG method can inherit the nonlinear stability of the continuous model. We apply a stabilization technique and prove unconditional energy stability of our scheme. Numerical experiments are performed to validate the analysis. Computational efficiency can be significantly enhanced by using this IMEX LDG method with a large time step.

High Order Implicit-explicit General Linear Methods with Optimized Stability Regions

In the numerical solution of partial differential equations using a method-of-lines approach, the availability of high order spatial discretization schemes motivates the development of sophisticated high order time integration methods. For multiphysics problems with both stiff and nonstiff terms implicit-explicit (IMEX) time stepping methods attempt to combine the lower cost advantage of explicit schemes with the favorable stability properties of implicit schemes. Existing high order IMEX Runge--Kutta or linear multistep methods, however, suffer from accuracy or stability limitations. This work shows that IMEX general linear methods (GLMs) are competitive alternatives to classic IMEX schemes for large problems arising in practice. High order IMEX-GLMs are constructed in the partitioned GLM framework developed earlier by the authors [J. Sci. Comput., 61 (2014), pp. 119--144]. The stability regions of the new schemes are optimized numerically. The resulting IMEX-GLMs have similar stability properties as IMEX Runge--Kutta methods, but they do not suffer from order reduction and are superior in terms of accuracy and efficiency. The new IMEX-GLMs have considerably better stability properties than the IMEX linear multistep methods. Numerical experiments with two- and three-dimensional test problems illustrate the potential of the new schemes to speed up complex applications.

Analysis of Asymptotic Preserving DG-IMEX Schemes for Linear Kinetic Transport Equations in a Diffusive Scaling

In this paper, some theoretical aspects will be addressed for the asymptotic preserving discontinuous Galerkin implicit-explicit (DG-IMEX) schemes recently proposed in [J. Jang, F. Li, J.-M. Qiu, and T. Xiong, High order asymptotic preserving DG-IMEX schemes for discrete-velocity kinetic equations in a diffusive scaling, http://arxiv.org/abs/1306.0227, 2013, submitted] for kinetic transport equations under a diffusive scaling. We will focus on the methods that are based on discontinuous Galerkin (DG) spatial discretizations with the $P^k$ polynomial space and a first order implicit-explicit (IMEX) temporal discretization, and apply them to two linear models: the telegraph equation and the one-group transport equation in slab geometry. In particular, we will establish uniform numerical stability with respect to Knudsen number $\varepsilon$ using energy methods, as well as error estimates for any given $\varepsilon$. When $\varepsilon\rightarrow 0$, a rigorous asymptotic analysis of the schemes is also obtained. Though the methods and the analysis are presented for one dimension in space, they can be generalized to higher dimensions directly.

A high order time-accurate loosely-coupled solution algorithm for unsteady conjugate heat transfer problems

Thermal interaction of fluids and solids, or conjugate heat transfer (CHT), is encountered in many engineering applications. Noting that time-accurate computations of transient CHT problems can be computationally expensive, we consider the use of high order implicit time integration schemes which have the potential to be computationally more efficient relative to the commonly used second order implicit schemes. For thermally weak couplings, we present a loosely-coupled solution algorithm where high order implicit–explicit (IMEX) Runge–Kutta schemes are employed for time integration. The IMEX schemes consist of the explicit first-stage singly diagonally implicit Runge–Kutta (ESDIRK) schemes, for advancing the solution in time within each separate fluid and solid subdomain, and the explicit Runge–Kutta (ERK) schemes, for explicit integration of part of the coupling terms. By considering a numerical example (an unsteady conjugate natural convection in an enclosure), temporal order preservation of the coupling algorithm (without subiterating) is demonstrated. In addition, the stability of the loosely-coupled algorithm is investigated numerically for the CHT test-case; when the ratio of the thermal effusivities of the fluid and solid subdomains is much smaller than unity, using large Fourier numbers of the subdomains is possible, indicating that time-step size is restricted by accuracy rather than stability. Furthermore, the (computational) work-(temporal) precision character of several time integration schemes in solving the CHT test-case is compared over a range of accuracy requirements; for time-accurate solutions, the fourth and fifth order IMEX schemes are 1.5 times more efficient than Crank–Nicolson and 2.7 times more efficient than BDF2. The computational gain is higher for smaller tolerances.

Higher-order relaxation schemes for hyperbolic systems of conservation laws

We present a higher order generalization for relaxation methods in the framework presented by Jin and Xin in [10]. The schemes employ general higher order integration for spatial discretization and higher order implicit-explicit (IMEX) schemes or Total Variation diminishing (TVD) Runge–Kutta schemes for time integration of relaxing or relaxed schemes, respectively, for time integration. Numerical experiments are performed on various test problems, in particular, the Burger's and Euler equations of inviscid gas dynamics in both one and two space dimensions. In addition, uniform convergence with respect to the relaxation parameter is demonstrated.

Higher-order relaxation schemes for hyperbolic systems of conservation laws

We present a higher order generalization for relaxation methods in the framework presented by Jin and Xin in [10]. The schemes employ general higher order integration for spatial discretization and higher order implicit-explicit (IMEX) schemes or Total Variation diminishing (TVD) Runge–Kutta schemes for time integration of relaxing or relaxed schemes, respectively, for time integration. Numerical experiments are performed on various test problems, in particular, the Burger's and Euler equations of inviscid gas dynamics in both one and two space dimensions. In addition, uniform convergence with respect to the relaxation parameter is demonstrated.