Implicit-explicit Runge-Kutta Methods Research Articles

Implicit-explicit Runge–Kutta methods for pricing financial derivatives in state-dependent regime-switching jump-diffusion models

Implicit-explicit Runge–Kutta methods for pricing financial derivatives in state-dependent regime-switching jump-diffusion models

Energy diminishing implicit-explicit Runge–Kutta methods for gradient flows

This study focuses on the development and analysis of a group of high-order implicit-explicit (IMEX) Runge–Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time-step size, thanks to a stabilization technique. The stabilization constants are solely dependent on the minimal eigenvalues that result from the Butcher tables of the IMEX-RKs. Furthermore, we establish a simple framework that can determine whether an IMEX-RK scheme is capable of preserving the original energy dissipation property or not. We also present a heuristic convergence analysis based on the truncation errors. This is the first research to prove that a linear high-order single-step scheme can ensure the original energy stability unconditionally for general gradient flows. Additionally, we provide several high-order IMEX-RK schemes that satisfy the established framework. Notably, we discovered a new four-stage third-order IMEX-RK scheme that reduces energy. Finally, we provide numerical examples to demonstrate the stability and accuracy properties of the proposed methods.

A semi-implicit finite volume method for the Exner model of sediment transport

The aim of this work is to construct efficient finite volume schemes for the numerical study of sediment transport in shallow water, in the framework of the Exner model [7,10]. In most cases, the velocity related to the sediment is much lower that the fluid velocity, which, in turn, may be much lower that the free-surface wave speed. Explicit methods that resolve all waves require small time steps due to the CFL stability restriction because of fast surface waves. Furthermore, if Rusanov flux is adopted, slow sediment waves may be affected by the large numerical diffusion. The objective of the present work is to drastically improve the efficiency in the computation of the evolution of the sediment by treating water waves implicitly, thus allowing much larger time steps than the one required by fully explicit schemes. The goal is reached by suitably semi-implicit schemes obtained by the use of implicit-explicit Runge-Kutta methods.

Unconditionally energy-stable linear convex splitting algorithm for the L2 quasicrystals

Quasicrystals have extensive applications in material sciences. In this article, we develop an unconditional energy-dissipation-preserving, temporally second-order accurate, and linear method for solving a L2-gradient flow-based quasicrystal model. A truncated technique is introduced to regularize the nonlinear terms so that their second derivative is upper bounded. Based on the idea of the convex splitting method, we split free energy into the difference between two convex functionals. The convexity provides an appropriate choice of linear stabilization parameter. A linear and temporally second-order accurate algorithm is constructed utilizing the implicit-explicit Runge–Kutta method. In each time step, we can analytically prove that the proposed scheme satisfies the energy dissipation law. The Fourier spectral method is used to perform spatial discretization. Through numerical experiments, the accuracy, energy stability, and capability of the proposed method for quasicrystals are verified.

Global high-order numerical schemes for the time evolution of the general relativistic radiation magneto-hydrodynamics equations

Modeling correctly the transport of neutrinos is crucial in some astrophysical scenarios such as core-collapse supernovae and binary neutron star mergers. In this paper, we focus on the truncated-moment formalism, considering only the first two moments (M1 scheme) within the grey approximation, which reduces Boltzmann seven-dimensional equation to a system of equations closely resembling the hydrodynamic ones. Solving the M1 scheme is still mathematically challenging, since it is necessary to model the radiation-matter interaction in regimes where the evolution equations become stiff and behave as an advection-diffusion problem. Here, we present different global, high-order time integration schemes based on Implicit-Explicit Runge-Kutta methods designed to overcome the time-step restriction caused by such behavior while allowing us to use the explicit Runge-Kutta commonly employed for the magneto-hydrodynamics and Einstein equations. Finally, we analyze their performance in several numerical tests.

Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation

In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. the L 2 L^2 norm of the numerical solution does not increase in time, under the time step condition τ ≤ F ( h / c , d / c 2 ) \tau \le \mathcal {F}(h/c, d/c^2) , with the convection coefficient c c , the diffusion coefficient d d , and the mesh size h h . The function F \mathcal {F} depends on the specific IMEX temporal method, the polynomial degree k k of the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomes τ ≲ h / c \tau \lesssim h/c in the convection-dominated regime and it becomes τ ≲ d / c 2 \tau \lesssim d/c^2 in the diffusion-dominated regime. The result is improved for a first order IMEX-LDG method. To complement the theoretical analysis, numerical experiments are further carried out, leading to slightly stricter time step conditions that can be used by practitioners. Uniform stability with respect to the strength of the convection and diffusion effects can especially be relevant to guide the choice of time step sizes in practice, e.g. when the convection-diffusion equations are convection-dominated in some sub-regions.

Charge diffusion in relativistic resistive second-order dissipative magnetohydrodynamics

We study charge diffusion in relativistic resistive second-order dissipative magnetohydrodynamics. In this theory, charge diffusion is not simply given by the standard Navier-Stokes form of Ohm's law, but by an evolution equation which ensures causality and stability. This, in turn, leads to transient effects in the charge diffusion current, the nature of which depends on the particular values of the electrical conductivity and the charge-diffusion relaxation time. The ensuing equations of motion are of so-called stiff character, which requires special care when solving them numerically. To this end, we specifically develop an implicit-explicit Runge-Kutta method for solving relativistic resistive second-order dissipative magnetohydrodynamics and subject it to various tests. We then study the system's evolution in a simplified 1+1-dimensional scenario for a heavy-ion collision, where matter and electromagnetic fields are assumed to be transversely homogeneous, and investigate the cases of an initially non-expanding fluid and a fluid initially expanding according to a Bjorken scaling flow. In the latter case, the scale invariance is broken by the ensuing self-consistent dynamics of matter and electromagnetic fields. However, the breaking becomes quantitatively important only if the electromagnetic fields are sufficiently strong. The breaking of scale invariance is larger for smaller values of the conductivity. Aspects of entropy production from charge diffusion currents and stability are also discussed.

Development of a Three-Dimensional Hydrodynamic Model Based on the Discontinuous Galerkin Method

Though the discontinuous Galerkin method is attracting more and more applications in many fields due to its local conservation, high-order accuracy, and flexibility for resolving complex geometries, only a few three-dimensional hydrodynamic models based on the discontinuous Galerkin method are present. In this study, a three-dimensional hydrodynamic model with a σ-coordinate system in the vertical direction is developed. This model is discretized in space using the discontinuous Galerkin method and advanced in time with the implicit-explicit Runge–Kutta method. Numerical tests indicate that the developed model is convergent and can obtain better results with a smaller computational time when a higher approximation order is adopted. Other tests with exact solutions also indicate that the developed model can well simulate the vertical circulation under the effect of surface wind stress and the flow under the combined effect of wind stress and Coriolis acceleration terms. The simulation results of tidal flow in part of Bohai Bay, China, indicate that the model can be used for the simulation of tidal wave motion in realistic situations.

New two-derivative implicit-explicit Runge-Kutta methods for stiff reaction-diffusion systems

Reaction-diffusion systems are extensively used in the mathematical modeling of biological and chemical systems to explain the Turing instability. Generally, reaction-diffusion systems are highly stiff in both reaction and diffusion terms. This paper discusses a new class of two-derivative implicit-explicit (IMEX) Runge-Kutta (RK) type methods for the numerical simulations of stiff reaction-diffusion systems. The present methods do not require numerical inversion of the coefficient matrix - computationally explicit. Stability properties of the developed methods are compared with the similar methods discussed in the literature. Moreover, accuracy and efficiency of the developed methods are validated by numerical simulations of spatiotemporal pattern formations for different reaction-diffusion systems, such as, phase-separation, Schnakenberg model and electrodeposition process.

Implicit-explicit relaxation Runge-Kutta methods: construction, analysis and applications to PDEs

Spatial discretizations of time-dependent partial differential equations usually result in a large system of semi-linear and stiff ordinary differential equations. Taking the structures into account, we develop a family of linearly implicit and high order accurate schemes for the time discretization, using the idea of implicit-explicit Runge-Kutta methods and the relaxation techniques. The proposed schemes are monotonicity-preserving/conservative for the original problems, while the previous linearized methods are usually not. We also discuss the linear stability and strong stability preserving (SSP) property of the new relaxation methods. Numerical experiments on several typical models are presented to confirm the effectiveness of the proposed methods.

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 assessment of implicit-explicit time integrators for the pseudo-spectral approximation of Boussinesq thermal convection in an annulus

We analyze the behavior of an ensemble of time integrators applied to the semi-discrete problem resulting from the spectral discretization of the equations describing Boussinesq thermal convection in a cylindrical annulus. The equations are cast in their vorticity-streamfunction formulation that yields a differential algebraic equation (DAE). The ensemble comprises 28 members: 4 implicit-explicit multistep schemes, 22 implicit-explicit Runge-Kutta (IMEX-RK) schemes, and 2 fully explicit schemes used for reference. The schemes whose theoretical order varies from 2 to 5 are assessed for 11 different physical setups that cover laminar and turbulent regimes. Multistep and order 2 IMEX-RK methods exhibit their expected order of convergence under all circumstances. IMEX-RK methods of higher-order show occasional order reduction that impacts both algebraic and differential field variables. We ascribe the order reduction to the stiffness of the problem at hand and, to a larger extent, the presence of the DAE. Using the popular Crank-Nicolson Adams-Bashforth of order 2 (CNAB2) integrator as reference, performance is defined by the ratio of maximum admissible time step to the cost of performing one iteration; the maximum admissible time step is determined by inspection of the time series of viscous dissipation within the system, which guarantees a physically acceptable solution. Relative performance is bounded between 0.5 and 1.5 across all studied configurations. Considering accuracy jointly with performance, we find that 6 schemes consistently outperform CNAB2, meaning that in addition to allowing for a more efficient calculation, the accuracy that they achieve at their operational, dissipation-based limit of stability yields a lower error. In our most turbulent setup, where the behavior of the methods is almost entirely dictated by their explicit component, 13 IMEX-RK integrators outperform CNAB2 in terms of accuracy and efficiency.

A compact Fourth-Order Implicit-Explicit Runge-Kutta Type Method for Solving Diffusive Lotka–Volterra System

This paper aims to developed a high-order and accurate method for the solution of one-dimensional Lotka-Volterra-diffusion with Numman boundary conditions. A fourth-order compact finite difference scheme for spatial part combined with implicit-explicit Runge Kutta scheme in temporal are proposed. Furthermore, boundary points are discretized by using a compact finite difference scheme in terms of fourth order accuracy. A key idea for proposed scheme is to take full advantage of method of line (MOL), this is consequently enabling us to use implicit-explicit Runge Kutta method, that are of fourth order in time. We constructed fourth order accuracy in both space and time and is unconditionally stable. This is consequently leading to a reduction in the computational cost of the scheme. Numerical experiments show that the combination of the compact finite difference with IMEX- RK methods give an accurate and reliable for solving the Lotka-Volterra-diffusion.

New optimized implicit-explicit Runge-Kutta methods with applications to the hyperbolic conservation laws

This paper discusses a new class of optimized implicit-explicit Runge-Kutta methods for the numerical solution of the dispersive and non-dispersive hyperbolic systems. Optimized implicit-explicit methods are formulated for the better stability and dispersion properties. Moreover, for present methods inversion of the coefficient matrices is not necessary, which makes these methods very attractive in terms of computational cost and complexity. To validate the efficiency of the developed methods, we have solved the one- and two-dimensional dispersive rotating shallow water equations and benchmark problems from acoustics. Computed solutions are also compared with the exact and experimental results available in the literature. The present methods compete well with the existing multi-stage time-integration methods in terms of accurately resolving the physical characteristics for the chosen problems. Furthermore, the computational costs of the proposed methods are significantly lower as compared to the four-stage, fourth-order explicit Runge-Kutta (RK4) method.

Error estimates of local discontinuous Galerkin method with implicit-explicit Runge Kutta for two-phase miscible flow in porous media

Two-phase miscible flow in porous media is described by a coupled system of nonlinear partial differential equations. In this paper, we develop and analyze the fully discrete local discontinuous Galerkin method coupled with the implicit-explicit Runge Kutta (IMEX-RK) method for the problems. Second-order time discretization scheme in two adjacent time layer can be obtained with the IMEX-RK method. For the temporal-spatial discretization schemes, we give the optimal convergence analysis for both the pressure and concentration in L2-norm. We present several numerical examples for two-phase miscible flow problem to demonstrate the effectiveness and robustness of our method.

Modified shallow water model for viscous fluids and positivity preserving numerical approximation

Shallow water equations are widely used in the simulation of those geophysical flows for which the flow horizontal length scale is much greater than the vertical one. Inspired by the example of lava flows, we consider here a modified model with an additional transport equation for a scalar quantity (e.g., temperature), and the derivation of the shallow water equations from depth-averaging the Navier-Stokes equations is presented. The assumption of constant vertical profiles for some of the model variables is relaxed allowing the presence of vertical profiles, and it follows that the non-linearity of the flux terms results in the introduction of appropriate shape coefficients.The space discretization of the resulting system of hyperbolic partial differential equations is obtained with a modified version of the finite volume central-upwind scheme introduced by Kurganov and Petrova in 2007. The time discretization is based on an implicit-explicit Runge-Kutta method which couples properly the hyperbolic part and the stiff source terms, avoiding the use of a very small time step; the use of complex arithmetic increases accuracy in the implicit treatment of stiff terms. The whole scheme is proved to preserve the positivity of flow thickness and the stationary steady-states.Some numerical experiments are performed to validate the proposed method and to show the incidence on the numerical solutions of shape coefficients introduced in the model.

A linear, high-order, and unconditionally energy stable scheme for the epitaxial thin film growth model without slope selection

The epitaxial thin film growth model without slope selection is the L2-gradient flow of energy with a logarithmic potential in terms of the gradient of a height function. A challenge to numerically solving the model is how to treat the nonlinear term to preserve energy stability without compromising accuracy and efficiency. To resolve this problem, we present a high-order energy stable scheme by placing the linear and nonlinear terms in the convex and concave parts, respectively, and employing the specially designed implicit–explicit Runge–Kutta method. As a result, our scheme is linear, high-order accurate in time, and unconditionally energy stable. We show analytically that the scheme is unconditionally uniquely solvable and energy stable. Numerical experiments are presented to demonstrate the accuracy, efficiency, and energy stability of the proposed scheme.

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.