Numerical integration of semidiscrete evolution systems

Methods and algorithms for integrating initial value systems are examined. Of particular interest is efficient and accurate numerical integration of systems of ordinary differential equations that arise on semidiscrete spatial differencing or finite element projection for evolution problems characterized by partial differential equations. Integration schemes for general systems are described. Stiff and oscillatory systems are considered and these motivate selection of specific types of algorithms for certain problem classes. For example, we show that Runge-Kutta methods with extended regions of stability are particularly efficient for moderately stiff dissipative systems derived from parabolic transport equations. The theoretical developments of an earlier paper [1] determine bounds on stiffness and stability and may be used to examine the stiff dissipative or oscillatory nature of the system qualitatively. Order control and stepsize adjustment in variable-order, variable-step algorithms are compared for several integrators applied to stiff and nonstiff initial-value systems arising from representative parabolic evolution problems.     

Stability and convergence analysis of implicit–explicit one-leg methods for stiff delay differential equations

The purpose of this paper is devoted to studying the implicit–explicit (IMEX) one-leg methods for stiff delay differential equations (DDEs) which can be split into the stiff and nonstiff parts. IMEX one-leg methods are composed of implicit one-leg methods for the stiff part and explicit one-leg methods for the nonstiff part. We prove that if the IMEX one-leg methods is consistent of order 2 for the ordinary differential equations, and the implicit one-leg method is A-stable, then the IMEX one-leg methods for stiff DDEs are stable and convergent with order 2. Some numerical examples are given to verify the validity of the obtained theoretical results and the effectiveness of the presented methods.     

Fourth-order runge-kutta schemes for fluid mechanics applications

Multiple high-order time-integration schemes are used to solve stiff test problems related to the Navier-Stokes (NS) equations. The primary objective is to determine whether high-order schemes can displace currently used second-order schemes on stiff NS and Reynolds averaged NS (RANS) problems, for a meaningful portion of the work-precision spectrum. Implicit-Explicit (IMEX) schemes are used on separable problems that naturally partition into stiff and nonstiff components. Non-separable problems are solved with fully implicit schemes, oftentimes the implicit portion of an IMEX scheme. The convection-diffusion-reaction (CDR) equations allow a term by term stiff/nonstiff partition that is often well suited for IMEX methods. Major variables in CDR converge at near design-order rates with all formulations, including the fourth-order IMEX additive Runge-Kutta (ARK₂) schemes that are susceptible to order reduction. The semi-implicit backward differentiation formulae and IMEX ARK₂ schemes are of comparable efficiency. Laminar and turbulent aerodynamic applications require fully implicit schemes, as they are not profitably partitioned. All schemes achieve design-order convergence rates on the laminar problem. The fourth-order explicit singly diagonally implicit Runge-Kutta (ESDIRK4) scheme is more efficient than the popular second-order backward differentiation formulae (BDF2) method. The BDF2 and fourth-order modified extended backward differentiation formulae (MEBDF4) schemes are of comparable efficiency on the turbulent problem. High precision requirements slightly favor the MEBDF4 scheme (greater than three significant digits). Significant order reduction plagues the ESDIRK4 scheme in the turbulent case. The magnitude of the order reduction varies with Reynolds number. Poor performance of the high-order methods can partially be attributed to poor solver performance. Huge time steps allowed by high-order formulations challenge the capabilities of algebraic solver technology.     

ATOMFT: solving ODEs and DAEs using Taylor series

Taylor series methods compute a solution to an initial value problem in ordinary differential equations by expanding each component of the solution in a long Taylor series. The series terms are generated recursively using the techniques of automatic differentiation. The ATOMFT system includes a translator to transform statements of the system of ODEs into a FORTRAN 77 object program that is compiled, linked with the ATOMFT runtime library, and run to solve the problem. We review the use of the ATOMFT system for nonstiff and stiff ODEs, the propagation of global errors, and applications to differential algebraic equations arising from certain control problems, to boundary value problems, to numerical quadrature, and to delay problems.     

GPG-stability of Runge-Kutta methods for generalized delay differential systems

The GP_G-stability of Runge-Kutta methods for the numerical solutions of the systems of delay differential equations is considered. The stability behaviour of implicit Runge-Kutta methods (IRK) is analyzed for the solution of the system of linear test equations with multiple delay terms. After an establishment of a sufficient condition for asymptotic stability of the solutions of the system, a criterion of numerical stability of IRK with the Lagrange interpolation process is given for any stepsize of the method.     

Singly-implicit Runge-Kutta methods for retarded and ordinary differential equations

A continuous singly-implicit Runge-Kutta method is implemented for stiff retarded differential equations. The choice of this implicit Runge-Kutta method is based on stability investigations of wide classes of interpolationintegration schemes. The numerical results show the effectiveness of these methods for both stiff ordinary and retarded differential equations.     

D-Convergence of general linear methods for stiff delay differential equations

This paper is concerned with the numerical solution of delay differential equations (DDEs). We focus on the error behaviour of general linear methods for stiff DDEs. A restricted type of interpolation procedure is considered for general linear methods. D-convergence properties of general linear methods with this interpolation procedure are investigated. Some extant results are unified.     

Fourth-Order Runge–Kutta Schemes for Fluid Mechanics Applications

Multiple high-order time-integration schemes are used to solve stiff test problems related to the Navier–Stokes (NS) equations. The primary objective is to determine whether high-order schemes can displace currently used second-order schemes on stiff NS and Reynolds averaged NS (RANS) problems, for a meaningful portion of the work-precision spectrum. Implicit–Explicit (IMEX) schemes are used on separable problems that naturally partition into stiff and nonstiff components. Non-separable problems are solved with fully implicit schemes, oftentimes the implicit portion of an IMEX scheme. The convection–diffusion-reaction (CDR) equations allow a term by term stiff/nonstiff partition that is often well suited for IMEX methods. Major variables in CDR converge at near design-order rates with all formulations, including the fourth-order IMEX additive Runge–Kutta (ARK₂) schemes that are susceptible to order reduction. The semi-implicit backward differentiation formulae and IMEX ARK₂ schemes are of comparable efficiency. Laminar and turbulent aerodynamic applications require fully implicit schemes, as they are not profitably partitioned. All schemes achieve design-order convergence rates on the laminar problem. The fourth-order explicit singly diagonally implicit Runge–Kutta (ESDIRK4) scheme is more efficient than the popular second-order backward differentiation formulae (BDF2) method. The BDF2 and fourth-order modified extended backward differentiation formulae (MEBDF4) schemes are of comparable efficiency on the turbulent problem. High precision requirements slightly favor the MEBDF4 scheme (greater than three significant digits). Significant order reduction plagues the ESDIRK4 scheme in the turbulent case. The magnitude of the order reduction varies with Reynolds number. Poor performance of the high-order methods can partially be attributed to poor solver performance. Huge time steps allowed by high-order formulations challenge the capabilities of algebraic solver technology.     

Numerical study of one-step explicit-implicit methods of L-equivalent stiffly accurate two-stage Runge-Kutta schemes

The paper describes one-step methods for numerical integration of the Cauchy problem for systems of ordinary differential equations free from iterations and coinciding on linear problems (autonomous and non-autonomous) with stiffly accurate implicit two-stage Runge-Kutta (RK) schemes. The numerical study of their accuracy is performed on stiff tests, i.e., the autonomous Kaps system and non-autonomous Protero-Robinson problem.     

Comparison Of Interpolations Used In Solving Delay Differential Equations By Runge-Kutta Method

Delay differential equations are solved using embedded Singly Diagonally Implicit Runge-Kutta (SDIRK) method (3,4) in (4,5). The approximation of the delay term is obtained by Newton divided difference interpolation and interpolation developed by In't Hout. Numerical results based on these two types of interpolation are compared. The stability polynomial and the stability regions of SDIRK method (2,2) using In't Hout interpolation for the delay term are presented.     

Second-order stabilized explicit Runge-Kutta methods for stiff problems

Stabilized Runge-Kutta methods (they have also been called Chebyshev-Runge-Kutta methods) are explicit methods with extended stability domains, usually along the negative real axis. They are easy to use (they do not require algebra routines) and are especially suited for MOL discretizations of two- and three-dimensional parabolic partial differential equations. Previous codes based on stabilized Runge-Kutta algorithms were tested with mildly stiff problems. In this paper we show that they have some difficulties to solve efficiently problems where the eigenvalues are very large in absolute value (over 10⁵). We also develop a new procedure to build this kind of algorithms and we derive second-order methods with up to 320 stages and good stability properties. These methods are efficient numerical integrators of very large stiff ordinary differential equations. Numerical experiments support the effectiveness of the new algorithms compared to well-known methods as RKC, ROCK2, DUMKA3 and ROCK4.     

Parallel-Iterated RK-Type PC Methods With Continuous Output Formulas * This work was partly supported by N.R.P.F.S.

This paper investigates parallel predictor-corrector iteration schemes (PC iteration schemes) based on collocation Runge-Kutta corrector methods (RK corrector methods) with continuous output formulas for solving nonstiff initial-value problems (IVPs) for systems of first-order differential equations. The resulting parallel-iterated RK-type PC methods are also provided with continuous output formulas. The continuous numerical approximations are used for predicting the stage values in the PC iteration processes. In this way, we obtain parallel PC methods with continuous output formulas and high-accurate predictions. Applications of the resulting parallel PC methods to a few widely-used test problems reveal that these new parallel PC methods are much more efficient when compared with the parallel and sequential explicit RK methods from the literature.     

Legendre-Gauss-Radau Collocation Method for Solving Initial Value Problems of First Order Ordinary Differential Equations

In this paper, we propose an efficient numerical integration process for initial value problems of first order ordinary differential equations, based on Legendre-Gauss-Radau interpolation, which is easy to be implemented and possesses the spectral accuracy. We also develop a multi-step version of this approach, which can be regarded as a specific implicit Legendre-Gauss-Radau Runge-Kutta method, with the global convergence and the spectral accuracy. Numerical results coincide well with the theoretical analysis and demonstrate the effectiveness of these approaches.     

Iterative regularization method for lidar remote sensing

In this paper we present an inversion algorithm for ill-posed problems arising in atmospheric remote sensing. The proposed method is an iterative Runge-Kutta type regularization method. Those methods are better well known for solving differential equations. We adapted them for solving inverse ill-posed problems. The numerical performances of the algorithm are studied by means of simulations concerning the retrieval of aerosol particle size distributions from lidar observations.     

Comparison of two algorithms for solving STIFF differential equations

Many algorithms are available for solving differential equations. Of these, two methods—GEAR and STIFF3, which were developed specifically for stiff differential equations, are compared based on their performance on five test problems. The performance criteria are both accuracy of the numerical solution and efficiency of the method. The results indicate that GEAR, although the older of the two methods, is the preferred algorithm, for stiff differential equations.     

Implementation of high-order implicit runge-kutta methods

In this paper, we develop single-Newton iterative schemes for the solution of the stage equations of some implicit Runge-Kutta methods such as the four-stage Gauss and Radau IIA methods and the five-stage Lobatto IIIA formula. We also compare the implementation cost of these schemes with the simplified-Newton iteration and we present some numerical experiments on some well-known stiff test problems to show that the proposed iterations are reliable and efficient.     

The positivity of low-order explicit Runge-Kutta schemes applied in splitting methods

Splitting methods are frequently used for the solution of large stiff initial value problems of ordinary differential equations with an additively split right-hand side function. Such systems arise, for instance, as method of lines discretizations of evolutionary partial differential equations in many applications. We consider the choice of explicit Runge-Kutta (RK) schemes in implicit-explicit splitting methods. Our main objective is the preservation of positivity in the numerical solution of linear and nonlinear positive problems while maintaining a sufficient degree of accuracy and computational efficiency. A three-stage second-order explicit RK method is proposed which has optimized positivity properties. This method compares well with standard s-stage explicit RK schemes of order s, s = 2, 3. It has advantages in the low accuracy range, and this range is interesting for an application in splitting methods. Numerical results are presented.     

A practical method to integrate some stiff systems

A compact, absolutely stable numerical method is presented to integrate stiff systems of pseudo-linear, ordinary, first-order differential equations, commonly found in the simulation of biological models. Solutions are stepwise approximated by a complete set of first order rational polynomials. Mass balance is preserved by the approximations. No matrix inversions are required. Besides being stable, the method is also convergent and can be used with deferred approximation to the limit h = 0. Comparisons between this method and the Stoer-Bullirsch algorithm and fourth-order Runge-Kutta method are presented.     

Solving stiff differential equations for simulation

Computer simulation of dynamic systems very often leads to the solution of a set of stiff ordinary differential equations. The solution of this set of equations involves the eigenvalues of its Jacobian matrix. The greater the spread in eigenvalues, the more time consuming the solutions become when existing numerical methods are employed. Extremely stiff differential equations can become a very serious problem for some systems, rendering accurate numerical solutions completely uneconomic. In this paper, we propose new techniques for solving extremely stiff systems of differential equations. These algorithms are based on a class of implicit Runge-Kutta procedure with complete error estimate. The new techniques are applied to solving mathematical models of the relaxation problem behind blast waves.     

Cahn-Hilliard方程的紧致指数时间差分法研究

我们将一种快速稳定的紧致指数时间差分法应用于求解Cahn-Hilliard方程,并采用多步逼近法和龙格库塔方法有效地解决了方程中的非线性项带来的稳定问题。通过与经典的半隐式欧拉方法对比,分别对不同体自由能模型和不同扩散迁移率下的相场方程求解进行收敛性测试,验证了算法的正确性和高效性。最后我们用提出的方法对Flory-Huggins模型的粗化率进行研究,得到了与理论预测值一致的结果。