Singly diagonally implicit runge‐kutta method for time‐dependent reaction‐diffusion equation

In this article, an efficient fourth‐order accurate numerical method based on Padé approximation in space and singly diagonally implicit Runge‐Kutta method in time is proposed to solve the time‐dependent one‐dimensional reaction‐diffusion equation. In this scheme, we first approximate the spatial derivative using the second‐order central finite difference then improve it to fourth‐order by applying Padé approximation. A three stage fourth‐order singly diagonally implicit Runge‐Kutta method is then used to solve the resulting system of ordinary differential equations. It is also shown that the scheme is unconditionally stable, and is suitable for stiff problems. Several numerical examples are solved by the scheme and the efficiency and accuracy of the new scheme are compared with two widely used high‐order compact finite difference methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1423–1441, 2011     

Numerical experiments on a hybrid WENO5 filter for shock‐capturing

In the present paper, a hybrid filter is introduced for high accurate numerical simulation of shock‐containing flows. The fourth‐order compact finite difference scheme is used for the spatial discretization and the third‐order Runge–Kutta scheme is used for the time integration. After each time‐step, the hybrid filter is applied on the results. The filter is composed of a linear sixth‐order filter and the dissipative part of a fifth‐order weighted essentially nonoscillatory scheme (WENO5). The classic WENO5 scheme and the WENO5 scheme with adaptive order (WENO5‐AO) are used to form the hybrid filter. Using a shock‐detecting sensor, the hybrid filter reduces to the linear sixth‐order filter in smooth regions for damping high frequency waves and reduces to the WENO5 filter at shocks in order to eliminate unwanted oscillations produced by the nondissipative spatial discretization method. The filter performance and accuracy of the results are examined through several test cases including the advection, Euler and Navier–Stokes equations. The results are compared with that of a hybrid second‐order filter and also that of the WENO5 and WENO5‐AO schemes.     

A class of multi‐step difference schemes by using Padé approximant

In this paper, we present a method for the construction of a class of multi‐step finite differences schemes for solving arbitrary order linear two‐point boundary value problems. The construction technique is based on Padé approximant. It is easy to derive multi‐step difference schemes, and it includes many existing schemes as its special cases. Numerical experiments show that the proposed schemes are flexible and convergent. Copyright © 2013 John Wiley & Sons, Ltd.     

Smoothing with positivity‐preserving Padé schemes for parabolic problems with nonsmooth data

We introduce a new class of higher order numerical schemes for parabolic partial differential equations that are more robust than the well‐known Rannacher schemes. The new family of algorithms utilizes diagonal Padé schemes combined with positivity‐preserving Padé schemes instead of first subdiagonal Padé schemes. We utilize a partial fraction decomposition to address problems with accuracy and computational efficiency in solving the higher order methods and to implement the algorithms in parallel. Optimal order convergence for nonsmooth data is proved for the case of a self‐adjoint operator in Hilbert space as well as in Banach space for the general case. Numerical experiments support the theorems, including examples in pricing options with nonsmooth payoff in financial mathematics. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Nonstandard discretizations of the generalized Nagumo reaction‐diffusion equation

A competitive nonstandard semi‐explicit finite‐difference method is constructed and used to obtain numerical solutions of the diffusion‐free generalized Nagumo equation. Qualitative stability analysis and numerical simulations show that this scheme is more robust in comparison to some standard explicit methods such as forward Euler and the fourth‐order Runge‐Kutta method (RK4). The nonstandard scheme is extended to construct a semi‐explicit and an implicit scheme to solve the full Nagumo reaction‐diffusion equation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 363–379, 2003.     

A unified discontinuous Galerkin framework for time integration

We introduce a new discontinuous Galerkin approach for time integration. On the basis of the method of weighted residual, numerical quadratures are employed in the finite element time discretization to account for general nonlinear ordinary differential equations. Many different conditions, including explicit, implicit, and symplectic conditions, are enforced for the test functions in the variational analysis to obtain desirable features of the resulting time‐stepping scheme. The proposed discontinuous Galerkin approach provides a unified framework to derive various time‐stepping schemes, such as low‐order one‐step methods, Runge–Kutta methods, and multistep methods. On the basis of the proposed framework, several explicit Runge–Kutta methods of different orders are constructed. The derivation of symplectic Runge–Kutta methods has also been realized. The proposed framework allows the optimization of new schemes in terms of several characteristics, such as accuracy, sparseness, and stability. The accuracy optimization is performed on the basis of an analytical form of the error estimation function for a linear test initial value problem. Schemes with higher formal order of accuracy are found to provide more accurate solutions. We have also explored the optimization potential of sparseness, which is related to the general compressive sensing in signal/imaging processing. Two critical dimensions of the stability region, that is, maximal intervals along the imaginary and negative real axes, are employed as the criteria for stability optimization. This gives the largest Courant–Friedrichs–Lewy time steps in solving hyperbolic and parabolic partial differential equations, respectively. Numerical experiments are conducted to validate the optimized time‐stepping schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

SERK2v2: A new second‐order stabilized explicit Runge‐Kutta method for stiff problems

Traditionally, explicit numerical algorithms have not been used with stiff ordinary differential equations (ODEs) due to their stability. Implicit schemes are usually very expensive when used to solve systems of ODEs with very large dimension. Stabilized Runge‐Kutta methods (also called Runge–Kutta–Chebyshev methods) were proposed to try to avoid these difficulties. The Runge–Kutta methods are explicit methods with extended stability domains, usually along the negative real axis. They can easily be applied to large problem classes with low memory demand, they do not require algebra routines or the solution of large and complicated systems of nonlinear equations, and they are especially suited for discretizations using the method of lines of two and three dimensional parabolic partial differential equations. In Martín‐Vaquero and Janssen [Comput Phys Commun 180 (2009), 1802–1810], we showed that previous codes based on stabilized Runge–Kutta algorithms have some difficulties in solving problems with very large eigenvalues and we derived a new code, SERK2, based on sixth‐order polynomials. Here, we develop a new method based on second‐order polynomials with up to 250 stages and good stability properties. These methods are efficient numerical integrators of very stiff ODEs. Numerical experiments with both smooth and nonsmooth data support the efficiency and accuracy of the new algorithms when compared to other well‐known second‐order methods such as RKC and ROCK2. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Operator time‐splitting techniques combined with quintic B‐spline collocation method for the generalized Rosenau–KdV equation

In this article, the generalized Rosenau–KdV equation is split into two subequations such that one is linear and the other is nonlinear. The resulting subequations with the prescribed initial and boundary conditions are numerically solved by the first order Lie–Trotter and the second‐order Strang time‐splitting techniques combined with the quintic B‐spline collocation by the help of the fourth order Runge–Kutta (RK‐4) method. To show the accuracy and reliability of the proposed techniques, two test problems having exact solutions are considered. The computed error norms L₂ and L_∞ with the conservative properties of the discrete mass Q(t) and energy E(t) are compared with those available in the literature. The convergence orders of both techniques have also been calculated. Moreover, the stability analyses of the numerical schemes are investigated.     

A fast singly diagonally implicit runge–kutta method for solving 1D unsteady convection‐diffusion equations

In this article, a fast singly diagonally implicit Runge–Kutta method is designed to solve unsteady one‐dimensional convection diffusion equations. We use a three point compact finite difference approximation for the spatial discretization and also a three‐stage singly diagonally implicit Runge–Kutta (RK) method for the temporal discretization. In particular, a formulation evaluating the boundary values assigned to the internal stages for the RK method is derived so that a phenomenon of the order of the reduction for the convergence does not occur. The proposed scheme not only has fourth‐order accuracy in both space and time variables but also is computationally efficient, requiring only a linear matrix solver for a tridiagonal matrix system. It is also shown that the proposed scheme is unconditionally stable and suitable for stiff problems. Several numerical examples are solved by the new scheme and the numerical efficiency and superiority of it are compared with the numerical results obtained by other methods in the literature. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 788–812, 2014     

High‐order finite difference schemes for numerical solutions of the generalized Burgers–Huxley equation

In this article, up to tenth‐order finite difference schemes are proposed to solve the generalized Burgers–Huxley equation. The schemes based on high‐order differences are presented using Taylor series expansion. To establish the numerical solutions of the corresponding equation, the high‐order schemes in space and a fourth‐order Runge‐Kutta scheme in time have been combined. Numerical experiments have been conducted to demonstrate the high‐order accuracy of the current algorithms with relatively minimal computational effort. The results showed that use of the present approaches in the simulation is very applicable for the solution of the generalized Burgers–Huxley equation. The current results are also seen to be more accurate than some results given in the literature. The proposed algorithms are seen to be very good alternatives to existing approaches for such physical applications. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1313‐1326, 2011     

Non‐perturbative solution of three‐dimensional Navier–Stokes equations for the flow near an infinite rotating disk

In this paper, we present Homotopy perturbation method (HPM) and Padé technique, for finding non‐perturbative solution of three‐dimensional viscous flow near an infinite rotating disk. We compared our solution with the numerical solution (fourth‐order Runge–Kutta). The results show that the HPM–Padé technique is an appropriate method in solving the systems of nonlinear equations. The mathematical technique employed in this paper is significant in studying some other problems of engineering. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical study of the asymptotics of the second Painlevé equation by a functional fitting method

The Painlevé equations arise as reductions of the soliton equations such as the Korteweg–de Vries equation, the nonlinear Schrödinger equation and so on. In this study, we are concerned with numerical approximation of the asymptotics of solutions of the second Painlevé equation on pole‐free intervals along the real axis. Classical integrators such as high order Runge–Kutta schemes might be expensive to simulate oscillation, decay and blow‐up behaviours depending on initial conditions. However, a lower order functional fitting method catches all kinds of solutions even for relatively large step sizes. Copyright © 2013 John Wiley & Sons, Ltd.     

Local discontinuous Galerkin methods with explicit Runge‐Kutta time marching for nonlinear carburizing model

A fully discrete local discontinuous Galerkin (LDG) method coupled with 3 total variation diminishing Runge‐Kutta time‐marching schemes, for solving a nonlinear carburizing model, will be analyzed and implemented in this paper. On the basis of a suitable numerical flux setting in the LDG method, we obtain the optimal error estimate for the Runge‐Kutta–LDG schemes by energy analysis, under the condition τ ≤ λh², where h and τ are mesh size and time step, respectively, λ is a positive constant independent of h. Numerical experiments are presented to verify the accuracy and capability of the proposed schemes. For the carburizing diffusion processes of steel and the diffusion simulation for Cu‐Ni system, the numerical results show good agreement with the experimental results.     

Runge–Kutta methods for jump–diffusion differential equations

In this paper we consider Runge–Kutta methods for jump–diffusion differential equations. We present a study of their mean-square convergence properties for problems with multiplicative noise. We are concerned with two classes of Runge–Kutta methods. First, we analyse schemes where the drift is approximated by a Runge–Kutta ansatz and the diffusion and jump part by a Maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the Runge–Kutta ansatz for the drift. The second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise. We present results showing when the implicit stochastic equations defining the stage values of the Runge–Kutta methods are uniquely solvable. Finally, simulation results illustrate the theoretical findings.     

Positively invariant cones of dynamical systems under Runge‐Kutta and Rosenbrock‐type discretization

In this paper we consider positively invariant cones of finite dimensional dynamical systems and study conditions on the time step‐size that guarantee the discrete positive invariance of these cones under Runge‐Kutta and Rosenbrock‐type methods. We conclude quite simple sufficient conditions, which involve the positivity (or absolute monotonicity) radius of the Runge‐Kutta schemes and its generalization when the Rosenbrock‐type methods are applied. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evolutionary generation of high‐order,explicit, two‐step methods for second‐order linear IVPs

In this paper, we consider the integration of systems of second‐order linear inhomogeneous initial value problems with constant coefficients. Hybrid Numerov methods are used that are constructed in the sense of Runge‐Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. We present the order conditions taking advantage of the special structure of the problem at hand. These equations are solved using differential evolution technique, and we present a method with algebraic order eighth at a cost of only 5 function evaluations per step. Numerical results over some linear problems, especially arising from the semidiscretization of the wave equation indicate the superiority of the new method.     

Rooted tree analysis of Runge–Kutta methods with exact treatment of linear terms

We investigate a class of time discretization schemes called “ETD Runge–Kutta methods,” where the linear terms of an ordinary differential equation are treated exactly, while the other terms are numerically integrated by a one-step method. These schemes, proposed by previous authors, can be regarded as modified Runge–Kutta methods whose coefficients are matrices instead of scalars. From this viewpoint, we reexamine the notion of consistency, convergence and order to provide a mathematical foundation for new methods. Applying the rooted tree analysis, expansion theorems of both the strict and numerical solutions are proved, and two types of order conditions are defined. Several classes of formulas with up to four stages that satisfy the conditions are constructed, and it is shown that the power series of matrices, employed as their coefficients, can be determined using the order conditions.     

Arbitrarily high-order accurate and energy-stable schemes for solving the conservative Allen–Cahn equation

In this paper, three high-order accurate and unconditionally energy-stable methods are proposed for solving the conservative Allen–Cahn equation with a space–time dependent Lagrange multiplier. One is developed based on an energy linearization Runge–Kutta (EL–RK) method which combines an energy linearization technique with a specific class of RK schemes, the other two are based on the Hamiltonian boundary value method (HBVM) including a Gauss collocation method, which is the particular instance of HBVM, and a general class of cases. The system is first discretized in time by these methods in which the property of unconditional energy stability is proved. Then the Fourier pseudo-spectral method is employed in space along with the proofs of mass conservation. To show the stability and validity of the obtained schemes, a number of 2D and 3D numerical simulations are presented for accurately calculating geometric features of the system. In addition, our numerical results are compared with other known structure-preserving methods in terms of numerical accuracy and conservation properties.     

Asymptotic preserving time‐discretization of optimal control problems for the Goldstein–Taylor model

We consider the development of implicit‐explicit time integration schemes for optimal control problems governed by the Goldstein–Taylor model. In the diffusive scaling, this model is a hyperbolic approximation to the heat equation. We investigate the relation of time integration schemes and the formal Chapman–Enskog‐type limiting procedure. For the class of stiffly accurate implicit–explicit Runge–Kutta methods, the discrete optimality system also provides a stable numerical method for optimal control problems governed by the heat equation. Numerical examples illustrate the expected behavior. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1770–1784, 2014     

A new family of 7 stages,eighth‐order explicit Numerov‐type methods

In this paper, we consider the integration of the special second‐order initial value problem. Hybrid Numerov methods are used, which are constructed in the sense of Runge‐Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. A new family of such methods attaining eighth algebraic order is given at a cost of only 7 function evaluations per step. The new family provides us with an extra parameter, which is used to increase phase‐lag order to 18. We proceed with numerical tests over a standard set of problems for these cases. Appendices implementing the symbolic construction of the methods and derivation of their coefficients are also given.