Implicit-explicit Runge-Kutta Methods Research Articles

Parallel Implicit-Explicit General Linear Methods

High-order discretizations of partial differential equations (PDEs) necessitate high-order time integration schemes capable of handling both stiff and nonstiff operators in an efficient manner. Implicit-explicit (IMEX) integration based on general linear methods (GLMs) offers an attractive solution due to their high stage and method order, as well as excellent stability properties. The IMEX characteristic allows stiff terms to be treated implicitly and nonstiff terms to be efficiently integrated explicitly. This work develops two systematic approaches for the development of IMEX GLMs of arbitrary order with stages that can be solved in parallel. The first approach is based on diagonally implicit multistage integration methods (DIMSIMs) of types 3 and 4. The second is a parallel generalization of IMEX Euler and has the interesting feature that the linear stability is independent of the order of accuracy. Numerical experiments confirm the theoretical rates of convergence and reveal that the new schemes are more efficient than serial IMEX GLMs and IMEX Runge-Kutta methods.

Boundary treatment of implicit-explicit Runge-Kutta method for hyperbolic systems with source terms

In this paper, we develop a high order finite difference boundary treatment method for the implicit-explicit (IMEX) Runge-Kutta (RK) schemes solving hyperbolic systems with possibly stiff source terms on a Cartesian mesh. The main challenge is how to obtain the solutions at ghost points resulting from the wide stencil of the interior high order scheme. We address this problem by combining the idea of using the RK schemes at the boundary and an inverse Lax-Wendroff (ILW) procedure in [29,30]. Specifically, we only apply the ILW procedure in the starting stage of the RK method. In the intermediate stages, we solve out the intermediate solutions as well as their first-order spatial derivatives at the boundary by using the RK schemes, which are then used to compute solutions at ghost points by Taylor expansions. Our method is different from the widely used approach for the explicit RK schemes by imposing boundary conditions at intermediate stages [14,29,30] which does not apply to IMEX schemes. In addition, the intermediate boundary conditions are available for explicit RK schemes only up to third order while our method applies to IMEX and explicit RK schemes of arbitrary order. The stability and the desired accuracy of our boundary treatment with IMEX RK method up to fourth order are demonstrated numerically through 1D examples and 2D reactive Euler equations.

An Efficient and Accurate Method for the Conservative Swift–Hohenberg Equation and Its Numerical Implementation

The conservative Swift–Hohenberg equation was introduced to reformulate the phase-field crystal model. A challenge in solving the conservative Swift–Hohenberg equation numerically is how to treat the nonlinear term to preserve mass conservation without compromising efficiency and accuracy. To resolve this problem, we present a linear, high-order, and mass conservative method by placing the linear and nonlinear terms in the implicit and explicit parts, respectively, and employing the implicit-explicit Runge–Kutta method. We show analytically that the method inherits the mass conservation. Numerical experiments are presented demonstrating the efficiency and accuracy of the proposed method. In particular, long time simulation for pattern formation in 2D is carried out, where the phase diagram can be observed clearly. The MATLAB code for numerical implementation of the proposed method is provided in Appendix.

Boundary treatment of high order Runge-Kutta methods for hyperbolic conservation laws

In [4], we developed a boundary treatment method for implicit-explicit (IMEX) Runge-Kutta (RK) methods for solving hyperbolic systems with source terms. Since IMEX RK methods include explicit ones as special cases, this boundary treatment method naturally applies to explicit methods as well. In this paper, we examine this boundary treatment method for the case of explicit RK schemes of arbitrary order applied to hyperbolic conservation laws. We show that the method not only preserves the accuracy of explicit RK schemes but also possesses good stability. This compares favorably to the inverse Lax-Wendroff method in [5,6] where analysis and numerical experiments have previously verified the presence of order reduction [5,6]. In addition, we demonstrate that our method performs well for strong-stability-preserving (SSP) RK schemes involving negative coefficients and downwind spatial discretizations. It is numerically shown that when boundary conditions are present and the proposed boundary treatment is used, that SSP RK schemes with negative coefficients still allow for larger time steps than schemes with all nonnegative coefficients. In this regard, our boundary treatment method is an effective supplement to SSP RK schemes with/without negative coefficients for initial-boundary value problems for hyperbolic conservation laws.

Stability-enhanced AP IMEX-LDG schemes for linear kinetic transport equations under a diffusive scaling

Transport equations arise in many applications such as rarefied gas dynamics, neutron transport, and radiative transfer. In this work, we consider some linear kinetic transport equations in a diffusive scaling and design high order asymptotic preserving (AP) methods within the discontinuous Galerkin method framework, with the main objective to achieve unconditional stability in the diffusive regime when the Knudsen number ε≪1, and to achieve high order accuracy when ε=O(1) and when ε≪1. Initial layers are also taken into account. The ingredients to accomplish our goal include: model reformulations based on the micro-macro decomposition and the limiting diffusive equation, local discontinuous Galerkin (LDG) methods in space, globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta methods in time, and strategies to handle non-well prepared initial data. Formal asymptotic analysis is carried out for the continuous model within the micro-macro decomposed framework to derive the initial layer as well as the interior problem with an asymptotically consistent initial condition as ε→0, and it is also conducted for numerical schemes to show the AP property and to understand the numerical initial treatments in the presence of initial layers. Fourier type stability analysis is performed, and it confirms the unconditional stability in the diffusive regime, and moreover it gives the stability condition in the kinetic regime when ε=O(1). In the reformulation step, a weighted diffusive term is added and subtracted to remove the parabolic stiffness and enhance the numerical stability in the diffusive regime. Such idea is not new, yet our numerical stability and asymptotic analysis provide new mathematical understanding towards the desired properties of the weight function. Finally, numerical examples are presented to demonstrate the accuracy, stability, and asymptotic preserving property of the proposed methods, as well as the effectiveness of the proposed strategies in the presence of the initial layer.

High-order partitioned spectral deferred correction solvers for multiphysics problems

We present an arbitrarily high-order, conditionally stable, partitioned spectral deferred correction (SDC) method for solving multiphysics problems using a sequence of pre-existing single-physics solvers. This method extends the work in [1,2], which used implicit-explicit Runge-Kutta methods (IMEX) to build high-order, partitioned multiphysics solvers. We consider a generic multiphysics problem modeled as a system of coupled ordinary differential equations (ODEs), coupled through coupling terms that can depend on the state of each subsystem; therefore the method applies to both a semi-discretized system of partial differential equations (PDEs) or problems naturally modeled as coupled systems of ODEs. The sufficient conditions to build arbitrarily high-order partitioned SDC schemes are derived. Based on these conditions, various of partitioned SDC schemes are designed. The stability of the first-order partitioned SDC scheme is analyzed in detail on a coupled, linear model problem. We show that the scheme is conditionally stable, and under conditions on the coupling strength, the scheme can be unconditionally stable. We demonstrate the performance of the proposed partitioned solvers on several classes of multiphysics problems with moderate coupling strength. They include a stiff linear system of ODEs, advection-diffusion-reaction systems, and fluid-structure interaction problems with both incompressible and compressible flows, where we verify the design order of the SDC schemes and study various stability properties. We also directly compare the accuracy, stability, and cost of the proposed partitioned SDC solver with the partitioned IMEX method in [1,2] on this suite of test problems. The results suggest that the high-order partitioned SDC solvers are more robust than the partitioned IMEX solvers for the numerical examples considered in this work, while the IMEX methods require fewer implicit solves.

Solving semi-linear stiff neutral equations by implicit–explicit Runge-Kutta methods

In this paper, for solving semi-linear stiff neutral equations, a class of extended implicit–explicit Runge-Kutta (EIERK) methods are suggested. Under some suitable conditions, it is shown that an EIERK method is convergent of order one or two. The presented numerical examples further verify the theoretical results and effectiveness of the methods. A comparison with the adapted fully implicit Runge-Kutta methods shows that EIERK methods possess the higher computational efficiency.

A High-Order Convex Splitting Method for a Non-Additive Cahn–Hilliard Energy Functional

Various Cahn–Hilliard (CH) energy functionals have been introduced to model phase separation in multi-component system. Mathematically consistent models have highly nonlinear terms linked together, thus it is not well-known how to split this type of energy. In this paper, we propose a new convex splitting and a constrained Convex Splitting (cCS) scheme based on the splitting. We show analytically that the cCS scheme is mass conserving and satisfies the partition of unity constraint at the next time level. It is uniquely solvable and energy stable. Furthermore, we combine the convex splitting with the specially designed implicit–explicit Runge–Kutta method to develop a high-order (up to third-order) cCS scheme for the multi-component CH system. We also show analytically that the high-order cCS scheme is unconditionally energy stable. Numerical experiments with ternary and quaternary systems are presented, demonstrating the accuracy, energy stability, and capability of the proposed high-order cCS scheme.

A parallel hp-adaptive high order discontinuous Galerkin method for the incompressible Navier-Stokes equations

We present a parallel hp-adaptive high order (spectral) discontinuous Galerkin method for approximation of the incompressible Navier-Stokes equations. The spatial discretization consists of equal-order polynomial approximations of the fluid velocity and pressure via discontinuous Galerkin spatial discretizations. For the nonlinear convective term we select the local Lax-Friedrichs flux, while for the divergence and gradient operators central fluxes are chosen. For the diffusive term, we use an interior penalty discontinuous Galerkin method to ensure stability and invertibility. The temporal discretization is an implicit-explicit Runge-Kutta method paired with a high-order splitting procedure to efficiently enforce the incompressibility condition at each time step. The compact stencil size, explicit time stepping of nonlinear terms, and inversion of sparse linear systems make the resulting method simple to parallelize while the local nature of the discontinuous Galerkin approximation makes hp-adaptive refinement natural to implement. We detail our implementation consisting of a tensor product basis of high order polynomials on quadrilateral elements, and implement hp-adaptivity using an inexpensive a posteriori error estimator to determine where refinement is necessary. p-Multigrid and pressure projection techniques are used to precondition the conjugate gradient linear solvers. We present several numerical tests to demonstrate the efficacy of the method, in particular in reducing the number of degrees of freedom needed and allocating computing resources to regions of sharp variation in transient incompressible Navier-Stokes flows.

Numerical solution of a spatio-temporal predator-prey model with infected prey.

A spatio-temporal eco-epidemiological model is formulated by combining an available non-spatial model for predator-prey dynamics with infected prey [D. Greenhalgh and M. Haque, Math. Meth. Appl. Sci., 30 (2007), 911-929] with a spatio-temporal susceptible-infective (SI)-type epidemic model of pattern formation due to diffusion [G.-Q. Sun, Nonlinear Dynamics, 69 (2012), 1097-1104]. It is assumed that predators exclusively eat infected prey, in agreement with the hypothesis that the infection weakens the prey, making it available for predation otherwise we assume that the predator has essentially no access to healthy prey of the same species. Furthermore, the movement of predators is described by a non-local convolution of the density of infected prey as proposed in [R.M. Colombo and E. Rossi, Commun. Math. Sci., 13 (2015), 369-400]. The resulting convection-diffusion-reaction system of three partial differential equations for the densities of susceptible and infected prey and predators is solved by an efficient method that combines weighted essentially non-oscillatory (WENO) reconstructions and an implicit-explicit Runge-Kutta (IMEX-RK) method for time stepping. Numerical examples illustrate the formation of spatial patterns involving all three species.

Superconvergent IMEX peer methods

In this paper we will focus on numerical methods for differential equations with both stiff and nonstiff parts. This kind of problems can be treated efficiently by implicit-explicit (IMEX) methods and here we investigate a class of s-stage IMEX peer methods of order p=s for variable and p=s+1 for constant step sizes. They are combinations of s-stage superconvergent implicit and explicit peer methods. We construct methods of order p=s+1 for s=3,4,5 where we compute the free parameters numerically to give good stability with respect to a general linear test problem frequently used in the literature. Numerical comparisons with two-step IMEX Runge–Kutta methods confirm the high potential of the new constructed superconvergent IMEX peer methods.

A Fully Implicit Finite Volume Lattice Boltzmann Method for Turbulent Flows

AbstractAlmost all schemes existed in the literature to solve the Lattice Boltzmann Equation likestream & collide, finite difference, finite element, finite volumeschemes are explicit. However, it is known fact that implicit methods utilizes better stability and faster convergence compared to the explicit methods. In this paper, a method named herein as Implicit Finite Volume Lattice Boltzmann Method (IFVLBM) for incompressible laminar and turbulent flows is proposed and it is applied to some 2D benchmark test cases given in the literature.Alternating Direction Implicit, an approximate factorization method is used to solve the obtained algebraic system. The proposed method presents a very good agreement for all the validation cases with the literature data. The proposed method shows good stability characteristics, the CFL number is eased. IFVLBM has about 2 times faster convergence rate compared with Implicit-Explicit Runge Kutta method even though it possesses a computational burden from the solution of algebraic systems of equations.

Numerical solution of a spatio-temporal gender-structured model for hantavirus infection in rodents.

In this article we describe the transmission dynamics of hantavirus in rodents using a spatio-temporal susceptible-exposed-infective-recovered (SEIR) compartmental model that distinguishes between male and female subpopulations [L.J.S. Allen, R.K. McCormack and C.B. Jonsson, Bull. Math. Biol. 68 (2006), 511--524]. Both subpopulations are assumed to differ in their movement with respect to local variations in the densities of their own and the opposite gender group. Three alternative models for the movement of the male individuals are examined. In some cases the movement is not only directed by the gradient of a density (as in the standard diffusive case), but also by a non-local convolution of density values as proposed, in another context, in [R.M. Colombo and E. Rossi, Commun. Math. Sci., 13 (2015), 369--400]. An efficient numerical method for the resulting convection-diffusion-reaction system of partial differential equations is proposed. This method involves techniques of weighted essentially non-oscillatory (WENO) reconstructions in combination with implicit-explicit Runge-Kutta (IMEX-RK) methods for time stepping. The numerical results demonstrate significant differences in the spatio-temporal behavior predicted by the different models, which suggest future research directions.

Convex Splitting Runge–Kutta methods for phase-field models

In this paper, we present the Convex Splitting Runge–Kutta (CSRK) methods which provide a simple unified framework to solve phase-field models such as the Allen–Cahn, Cahn–Hilliard, and phase-field crystal equations. The core idea of the CSRK methods is the combination of convex splitting methods and multi-stage implicit–explicit Runge–Kutta methods. Our CSRK methods are high-order accurate in time and we investigate the energy stability numerically. We present numerical experiments to show the accuracy and efficiency of the proposed methods up to the third-order accuracy.

A second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin scheme for the Kerr–Debye model

In this paper, we develop a second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) scheme for the Kerr–Debye model. By using the approach first introduced by Zhang and Shu in [Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal. 42 (2004) 641–666.] with an energy estimate and Taylor expansion, the asymptotic-preserving property of the semi-discrete DG methods is proved rigorously. In addition, we propose a class of unconditional positivity-preserving implicit–explicit (IMEX) Runge–Kutta methods for the system of ordinary differential equations arising from the semi-discretization of the Kerr–Debye model. The new IMEX Runge–Kutta methods are based on the modification of the strong-stability-preserving (SSP) implicit Runge–Kutta method and have second-order accuracy. The numerical results validate our analysis.

A Unified IMEX Runge--Kutta Approach for Hyperbolic Systems with Multiscale Relaxation

In this paper we consider the development of Implicit-Explicit (IMEX) Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such systems the scaling depends on an additional parameter which modifies the nature of the asymptotic behavior which can be either hyperbolic or parabolic. Because of the multiple scalings, standard IMEX Runge-Kutta methods for hyperbolic systems with relaxation loose their efficiency and a different approach should be adopted to guarantee asymptotic preservation in stiff regimes. We show that the proposed approach is capable to capture the correct asymptotic limit of the system independently of the scaling used. Several numerical examples confirm our theoretical analysis.

Highly stable implicit–explicit Runge–Kutta methods

We investigate implicit–explicit (IMEX) Runge–Kutta (RK) methods for differential systems with non-stiff and stiff processes. The construction of such methods with large regions of absolute stability of the ‘explicit part’ of the method assuming that the ‘implicit part’ of the scheme is A-stable, is described. We also describe the construction of IMEX RK methods, where the ‘explicit part’ of the schemes have strong stability properties. Examples of highly stable IMEX RK methods are provided up to the order p=4. Numerical examples are also given which illustrate good performance of these schemes.

EB Scheme-Based Hybrid SE-FE DGTD Method for Multiscale EM Simulations

This communication presents an EB scheme subdomainlevel discontinuous Galerkin time domain (DGTD) method for multiscale simulations. It is an extension of the previous subdomain-level DGTD research by combining the degree of freedom efficiency of spectral element time domain method and the mesh flexibility of the finite element time domain method. Thus, the multiscale problems can be solved efficiently by separating the geometrically fine and coarse parts and meshing them with hexahedrons and tetrahedrons, respectively, via a nonconformal mesh. The implicit–explicit Runge-Kutta method is applied to the EB scheme DGTD method to obtain an efficient time integration approach.

On linearly implicit IMEX Runge-Kutta methods for degenerate convection-diffusion problems modeling polydisperse sedimentation

Implicit-explicit (IMEX) Runge-Kutta (RK) methods are suitable for the solution of nonlinear, possibly strongly degenerate, convection-diffusion problems, since the stability restrictions, coming from the explicitly treated convective part, are much less severe than those that would be deduced from an explicit treatment of the diffusive term. A particularly efficient variant of these schemes, so-called linearly implicit IMEX-RK schemes, arise from discretizing the diffusion terms in a way that more carefully distinguishes between stiff and nonstiff dependence, such that in each time step only a linear system needs to be solved. These schemes provide an efficient tool for the numerical exploration of sediment formation and composition under a strongly degenerate polydisperse sedimentation model.

Construction of Additive Semi-Implicit Runge–Kutta Methods with Low-Storage Requirements

Space discretization of some time-dependent partial differential equations gives rise to systems of ordinary differential equations in additive form whose terms have different stiffness properties. In these cases, implicit methods should be used to integrate the stiff terms while efficient explicit methods can be used for the non-stiff part of the problem. However, for systems with a large number of equations, memory storage requirement is also an important issue. When the high dimension of the problem compromises the computer memory capacity, it is important to incorporate low memory usage to some other properties of the scheme. In this paper we consider Additive Semi-Implicit Runge-Kutta (ASIRK) methods, a class of implicitexplicit Runge-Kutta methods for additive differential systems. We construct two second order 3-stage ASIRK schemes with low-storage requirements. Having in mind problems with stiffness parameters, besides accuracy and stability properties, we also impose stiff accuracy conditions. The numerical experiments done show the advantages of the new methods.