Explicit Runge-Kutta Scheme Research Articles

Unconditionally positive and conservative third order modified Patankar–Runge–Kutta discretizations of production–destruction systems

Modified Patankar–Runge–Kutta (MPRK) schemes are numerical methods for the solution of positive and conservative production–destruction systems. They adapt explicit Runge–Kutta schemes to ensure positivity and conservation irrespective of the time step size. The first two members of this class, the first order MPE scheme and the second order MPRK22(1) scheme, have been successfully applied in a large number of applications. Recently, a general definition of MPRK schemes was introduced and necessary as well as sufficient conditions to obtain first and second order MPRK schemes were presented. In this paper we derive necessary and sufficient conditions for third order MPRK schemes and introduce the first family of such schemes. The theoretical results are confirmed by numerical experiments considering linear and nonlinear as well as nonstiff and stiff systems of differential equations.

Study of flow over object problems by a nodal discontinuous Galerkin-lattice Boltzmann method

The flow over object problems are studied by a nodal discontinuous Galerkin-lattice Boltzmann method (NDG-LBM) in this work. Different from the standard lattice Boltzmann method, the current method applies the nodal discontinuous Galerkin method into the streaming process in LBM to solve the resultant pure convection equation, in which the spatial discretization is completed on unstructured grids and the low-storage explicit Runge-Kutta scheme is used for time marching. The present method then overcomes the disadvantage of standard LBM for depending on the uniform meshes. Moreover, the collision process in the LBM is completed by using the multiple-relaxation-time scheme. After the validation of the NDG-LBM by simulating the lid-driven cavity flow, the simulations of flows over a fixed circular cylinder, a stationary airfoil and rotating-stationary cylinders are performed. Good agreement of present results with previous results is achieved, which indicates that the current NDG-LBM is accurate and effective for flow over object problems.

Discontinuous Galerkin Finite Element Investigation on the Fully-Compressible Navier–Stokes Equations for Microscale Shock-Channels

Microfluidics is a multidisciplinary area founding applications in several fields such as the aerospace industry. Microelectromechanical systems (MEMS) are mainly adopted for flow control, micropower generation and for life support and environmental control for space applications. Microflows are modeled relying on both a continuum and molecular approach. In this paper, the compressible Navier–Stokes (CNS) equations have been adopted to solve a two-dimensional unsteady flow for a viscous micro shock-channel problem. In microflows context, as for the most gas dynamics applications, the CNS equations are usually discretized in space using finite volume method (FVM). In the present paper, the PDEs are discretized with the nodal discontinuous Galerkin finite element method (DG–FEM) in order to understand how the method performs at microscale level for compressible flows. Validation is performed through a benchmark test problem for microscale applications. The error norms, order of accuracy and computational cost are investigated in a grid refinement study, showing a good agreement and increasing accuracy with reference data as the mesh is refined. The effects of different explicit Runge–Kutta schemes and of different time step sizes have also been studied. We found that the choice of the temporal scheme does not really affect the accuracy of the numerical results.

Explicit Large Time Stepping with Third-Order Exponential Time Differencing Scheme for Hypersonic Chemical Non-Equilibrium Flow

In this paper, a third-order exponential time differencing scheme, named ETDRK3, was investigated for large time stepping in the computation of hypersonic non-equilibrium flow. The second-order Harten-TVD scheme was used for the spatial discretization. The efficient implementation of the scheme with diagonalization of Jacobin matrix was established and carried out for the semi-cylindrical around flow. Current observations showed that the numerical results were in good agreement with those obtained by the classical explicit three-stage Runge-Kutta scheme (RK3) and implicit LU scheme. Efficiency assessments promised the effectiveness of the ETDRK3 scheme. The rationality of the application of this scheme was proved by its preferable accuracy and efficiency.

Numerical Simulation of Dam Break Flows Using a Radial Basis Function Meshless Method with Artificial Viscosity

A simple radial basis function (RBF) meshless method is used to solve the two-dimensional shallow water equations (SWEs) for simulation of dam break flows over irregular, frictional topography involving wetting and drying. At first, we construct the RBF interpolation corresponding to space derivative operators. Next, we obtain numerical schemes to solve the SWEs, by using the gradient of the interpolant to approximate the spatial derivative of the differential equation and a third-order explicit Runge–Kutta scheme to approximate the temporal derivative of the differential equation. For the problems involving shock or discontinuity solutions, we use an artificial viscosity for shock capturing. Then, we apply our scheme for several theoretical two-dimensional numerical experiments involving dam break flows over nonuniform beds and moving wet-dry fronts over irregular bed topography. Promising results are obtained.

Efficient Explicit Time Stepping of High Order Discontinuous Galerkin Schemes for Waves

This work presents algorithms for the efficient implementation of discontinuous Galerkin methods with explicit time stepping for acoustic wave propagation on unstructured meshes of quadrilaterals or hexahedra. A crucial step towards efficiency is to evaluate operators in a matrix-free way with sum-factorization kernels. The method allows for general curved geometries and variable coefficients. Temporal discretization is carried out by low-storage explicit Runge-Kutta schemes and the arbitrary derivative (ADER) method. For ADER, we propose a flexible basis change approach that combines cheap face integrals with cell evaluation using collocated nodes and quadrature points. Additionally, a degree reduction for the optimized cell evaluation is presented to decrease the computational cost when evaluating higher order spatial derivatives as required in ADER time stepping. We analyze and compare the performance of state-of-the-art Runge-Kutta schemes and ADER time stepping with the proposed optimizations. ADER involves fewer operations and additionally reaches higher throughput by higher arithmetic intensities and hence decreases the required computational time significantly. Comparison of Runge-Kutta and ADER at their respective CFL stability limit renders ADER especially beneficial for higher orders when the Butcher barrier implies an overproportional amount of stages. Moreover, vector updates in explicit Runge--Kutta schemes are shown to take a substantial amount of the computational time due to their memory intensity.

Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics

We consider a smooth planar system having slow-fast motion, where the slow motion takes place near a curve γ. We explore the idea of replacing the original smooth system with a system with discontinuous right-hand side (DRHS system for short), whereby the DRHS system coincides with the smooth one away from a neighborhood of γ. After this reformulation, in the region of phase-space where γ is attracting for the DRHS system, we will obtain sliding motion on γ and numerical methods apt at integrating for sliding motion can be applied. Moreover, we further bypass resolving the sliding motion and monitor entries (transversal) and exits (tangential) on the curve γ, a fact that can be done independently of resolving for the motion itself. The end result is a method free from the need to adopt stiff integrators or to worry about resolving sliding motion for the DRHS system. We illustrate the performance of our method on a few problems, highlighting the feasibility of using simple explicit Runge-Kutta schemes, and that we obtain much the same orbits of the original smooth system.

Entropy Production by Explicit Runge–Kutta Schemes

We propose and test a formula for the separate computation of the temporal and spatial entropy production of fully discrete, finite-volume, explicit Runge–Kutta discretizations of systems of conservation laws.

$$L_2$$ L 2 Stability of Explicit Runge–Kutta Schemes

Explicit Runge–Kutta methods are standard tools in the numerical solution of ordinary differential equations (ODEs). Applying the method of lines to partial differential equations, spatial semidiscretisations result in large systems of ODEs that are solved subsequently. However, stability investigations of high-order methods for transport equations are often conducted only in the semidiscrete setting. Here, strong-stability of semidiscretisations for linear transport equations, resulting in ODEs with semibounded operators, are investigated. For the first time, it is proved that the fourth-order, ten-stage SSP method of Ketcheson (SIAM J Sci Comput 30(4):2113–2136, 2008) is strongly stable for general semibounded operators. Additionally, insights into fourth-order methods with fewer stages are presented.

Steepest descent optimisation of Runge–Kutta coefficients for second order implicit finite volume CFD codes

One of the key research topics in the computational fluid dynamics community is to improve the computational efficiency of steady-state finite volume codes. Real-world use cases require the solution to the Navier–Stokes equations for a wide range of Mach numbers, Reynolds numbers and mesh cell aspect ratios. This introduces stiffness in the discretised equations and therefore a slowdown in convergence. The community has pursued in particular two avenues to speed up the convergence of the corresponding error modes: Optimisation of Runge–Kutta coefficients for explicit Runge–Kutta schemes; and the introduction of implicit preconditioners, with a limited investigation of Runge–Kutta coefficients suitable to those implicit preconditioners. After proposing improvements to the implicit preconditioner, the present work proposes an optimisation procedure allowing the optimisation of the Runge–Kutta coefficients specifically for the implicit preconditioner. Employed on a realistic use case, the Runge–Kutta coefficients extracted with this method show a 20%–38% reduction of the number of iterations needed for convergence compared to Runge–Kutta coefficients recommended in the literature for comparable schemes and with the same computational cost per iteration.

An accurate predictor-corrector time integration method for structural dynamics

In this paper, a new accurate predictor-corrector time integration method is proposed, which has an appropriate stability domain. The dispersion error of the suggested approach is considerably smaller than that of the central difference scheme. Moreover, the presented technique introduces effectively numerical damping, which can be used to remove the spurious highfrequency components from the numerical responses. The authors’ approach has second-order accuracy for structures with and without damping. It is worth mentioning that the scheme solves the nonlinear problems without using iterative processes. The new algorithm is tested by using several linear and nonlinear numerical examples, and the outcomes are compared against the results of the central difference and explicit second-order Runge-Kutta schemes. The findings confirm the high accuracy and good performance of the suggested technique.

On a Class of Implicit–Explicit Runge–Kutta Schemes for Stiff Kinetic Equations Preserving the Navier–Stokes Limit

Implicit–explicit (IMEX) Runge–Kutta (RK) schemes are popular high order time discretization methods for solving stiff kinetic equations. As opposed to the compressible Euler limit (leading order asymptotics of the Boltzmann equation as the Knudsen number $$\varepsilon $$ goes to zero), their asymptotic behavior at the Navier–Stokes (NS) level (next order asymptotics) was rarely studied. In this paper, we analyze a class of existing IMEX RK schemes and show that, under suitable initial conditions, they can capture the NS limit without resolving the small parameter $$\varepsilon $$ , i.e., $$\varepsilon =o(\Delta t)$$ , $$\Delta t^m=o(\varepsilon )$$ , where m is the order of the explicit RK part in the IMEX scheme. Extensive numerical tests for BGK and ES-BGK models are performed to verify our theoretical results.

A combined method for solving 2D incompressible flow and heat transfer by spectral collocation method and artificial compressibility method

A combined method is developed for solving 2D steady incompressible flow and heat transfer by spectral collocation method (SCM) and artificial compressibility method (ACM). Instead of the steady governing equations, the unsteady artificial compressibility equations are introduced and solved. The steady results can be obtained by solving the unsteady equations when the pressure, velocity and temperature time derivatives approach to zero. The partial differential equations are discretized by SCM with Chebyshev polynomial and the spatial domain is discretized by the Chebyshev-Gauss-Lobbatto (CGL) collocation points. The artificial compressibility parameter c is investigated and determined firstly. Then, the accuracy of the SCM-ACM is tested by solving an exact solution with three time schemes (explicit scheme, implicit scheme and fourth order explicit Runge-Kutta scheme). Lastly, two classical cases (the lid-driven cavity flow and natural convection in a square cavity) are solved by the SCM-ACM with explicit scheme. The results show that the SCM-ACM can be used for the solving of incompressible flow and heat transfer, and high accuracy can be achieved after some calculation steps by the SCM-ACM especially with the fourth order explicit Runge-Kutta scheme. The SCM-ACM inherits the characteristics of exponential convergence and high accuracy from spectral method, and it is simplified, efficient and easy to implement which derives from the ACM. The SCM-ACM provides a new selection for the solution of incompressible flow and heat transfer.

Method of moving frames to solve the shallow water equations on arbitrary rotating curved surfaces

A novel numerical scheme is proposed to solve the shallow water equations (SWEs) on arbitrary rotating curved surfaces. Based on the method of moving frames (MMF) in which the geometry is represented by orthonormal vectors, the proposed scheme not only has the fewest dimensionality both in space and time, but also does not require either of metric tensors, composite meshes, or the ambient space. The MMF–SWE formulation is numerically discretized using the discontinuous Galerkin method of arbitrary polynomial order p in space and an explicit Runge–Kutta scheme in time. The numerical model is validated against six standard tests on the sphere and the optimal order of convergence of p+1 is numerically demonstrated. The MMF–SWE scheme is also demonstrated for its efficiency and stability on the general rotating surfaces such as ellipsoid, irregular, and non-convex surfaces.

Explicit Runge–Kutta schemes for incompressible flow with improved energy-conservation properties

The application of pseudo-symplectic Runge–Kutta methods to the incompressible Navier–Stokes equations is discussed in this work. In contrast to fully energy-conserving, implicit methods, these are explicit schemes of order p that preserve kinetic energy to order q, with q>p. Use of explicit methods with improved energy-conservation properties is appealing for convection-dominated problems, especially in case of direct and large-eddy simulation of turbulent flows. A number of pseudo-symplectic methods are constructed for application to the incompressible Navier–Stokes equations and compared in terms of accuracy and efficiency by means of numerical simulations.

Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations

AbstractIn this paper we develop explicit fast exponential Runge-Kutta methods for the numerical solutions of a class of parabolic equations. By incorporating the linear splitting technique into the explicit exponential Runge-Kutta schemes, we are able to greatly improve the numerical stability. The proposed numerical methods could be fast implemented through use of decompositions of compact spatial difference operators on a regular mesh together with discrete fast Fourier transform techniques. The exponential Runge-Kutta schemes are easy to be adopted in adaptive temporal approximations with variable time step sizes, as well as applied to stiff nonlinearity and boundary conditions of different types. Linear stabilities of the proposed schemes and their comparison with other schemes are presented. We also numerically demonstrate accuracy, stability and robustness of the proposed method through some typical model problems.

High order numerical methods for networks of hyperbolic conservation laws coupled with ODEs and lumped parameter models

In this paper we construct high order finite volume schemes on networks of hyperbolic conservation laws with coupling conditions involving ODEs. We consider two generalized Riemann solvers at the junction, one of Toro–Castro type and a solver of Harten, Enquist, Osher, Chakravarthy type. The ODE is treated with a Taylor method or an explicit Runge–Kutta scheme, respectively. Both resulting high order methods conserve quantities exactly if the conservation is part of the coupling conditions. Furthermore we present a technique to incorporate lumped parameter models, which arise from simplifying parts of a network. The high order convergence and the robust capturing of shocks are investigated numerically in several test cases.

High-order implicit residual smoothing time scheme for direct and large eddy simulations of compressible flows

Restrictions on the maximum allowable time step of explicit time integration methods for direct and large eddy simulations of compressible turbulent flows at high Reynolds numbers can be very severe, because of the extremely small space steps used close to solid walls to capture tiny and elongated boundary layer structures. A way of increasing stability limits is to use implicit time integration schemes. However, the price to pay is a higher computational cost per time step, higher discretization errors and lower parallel scalability. In quest for an implicit time scheme for scale-resolving simulations providing the best possible compromise between these opposite requirements, we develop a Runge–Kutta implicit residual smoothing (IRS) scheme of fourth-order accuracy, based on a bilaplacian operator. The implicit operator involves the inversion of scalar pentadiagonal systems, for which efficient parallel algorithms are available. The proposed method is assessed against two explicit and two implicit time integration techniques in terms of computational cost required to achieve a threshold level of accuracy. Precisely, the proposed time scheme is compared to four-stages and six-stages low-storage Runge–Kutta method, to the second-order IRS and to a second-order backward scheme solved by means of matrix-free quasi-exact Newton subiterations. Numerical results show that the proposed IRS scheme leads to reductions in computational time by a factor 3 to 5 for an accuracy comparable to that of the corresponding explicit Runge–Kutta scheme.

High-order and mass conservative methods for the conservative Allen–Cahn equation

The conservative Allen–Cahn (AC) equation has been studied analytically and numerically. Our mathematical analysis and numerical experiment, however, show that previous numerical methods are not second-order accurate in time and/or do not conserve the initial mass. The aim of this paper is to propose high-order and mass conservative methods for solving the conservative AC equation. In the methods, we discretize the conservative AC equation by using a Fourier spectral method in space and first-, second-, and third-order implicit–explicit Runge–Kutta schemes in time. We show that the methods inherit the mass conservation. Numerical experiments are presented demonstrating the accuracy and efficiency of proposed methods.

A second order residual based predictor–corrector approach for time dependent pollutant transport

We present a second order residual distribution scheme for scalar transport problems in shallow water flows. The scheme, suitable for the unsteady cases, is obtained adapting to the shallow water context the explicit Runge-Kutta schemes for scalar equations 1. The resulting scheme is decoupled from the hydrodynamics yet the continuity equation has to be considered in order to respect some important numerical properties at discrete level. Beyond the classical characteristics of the residual formulation presented in 1,2, we introduce the possibility to iterate the corrector step in order to improve the accuracy of the scheme. Another novelty is that the scheme is based on a precise monotonicity condition which guarantees the respect of the maximum principle. We thus end up with a scheme which is mass conservative, second order accurate and monotone. These properties are checked in the numerical tests, where the proposed approach is also compared to some finite volume schemes on unstructured grids. The results obtained show the interest in adopting the predictor-corrector scheme for pollutant transport applications, where conservation of the mass, monotonicity and accuracy are the most relevant concerns.