Second-order Explicit Runge-Kutta Research Articles

Efficient partitioned numerical integrators for myocardial cell models

Cardiac simulations are often performed by numerically solving the bidomain or monodomain equations. In these simulations, much of the computational workload is devoted to the time integration of the systems of ordinary differential equations that arise from the myocardial cell models. For such time integration, a well-established top choice is the Rush–Larsen (RL) method, which partitions the myocardial cell models according to the gating and non-gating equations and treats them with the exponential Euler method and the forward Euler method, respectively. The partitioning strategy of the RL method is based on previous work by Moore and Ramon, with the main difference being that the Moore–Ramon (MR) method integrates the non-gating equations with Heun’s method, a two-stage, second-order explicit Runge–Kutta (ERK2) method. Although the RL method is less computationally expensive per step than the MR method, it should not be a foregone conclusion that the RL is more efficient. In this paper, we demonstrate that in the MR method typically outperforms the RL method in terms of efficiency. Furthermore, two new families of numerical methods are proposed based on the same partitioning strategy but integrating the non-gating equations with other ERK2 methods and multi-stage, first-order Runge–Kutta–Chebyshev methods, respectively. We show that the RL method is outperformed on 34 out of 36 cell models tested.

Numerical solutions of the coupled unsteady nonlinear convection-diffusion equations based on generalized finite difference method

In this paper, the meshless Generalized Finite Difference Method (GFDM) in conjunction with the second-order explicit Runge-Kutta method (RK2 method) is presented to solve coupled unsteady nonlinear convection-diffusion equations (CDEs). Compared with the conventional Euler method, the RK2 method not only has higher accuracy but also reduces the possibility of numerical oscillation in time discretization, especially for the nonlinear and coupled cases. The generalized finite difference method, which is a localized collocation method, is famous for its simplicity and adaptability in the numerical solution of partial differential equations. Benefiting from Taylor series and moving least squares, its partial derivatives can be formed by a series of surrounding space points. In comparison with traditional finite difference methods, the proposed GFDM is free of mesh and available for irregular discretization nodes. In this study, the stencil selection algorithms are introduced to choose the stencil support of a certain node from the whole discretization nodes. Error analysis and numerical investigations are presented to demonstrate the effectiveness of the proposed GFDM for solving the coupled linear and nonlinear unsteady convection-diffusion equations. Then it is successfully applied to three benchmark examples of the coupled unsteady nonlinear CDEs encountered in the double-diffusive natural convection process, chemotaxis-haptotaxis model of cancer invasion, and thermo-hygro coupling model of concrete.

Linear Instability of the Fifth-Order WENO Method

The weighted essentially nonoscillatory (WENO) methods are popular spatial discretization methods for hyperbolic partial differential equations. In this paper we show that the combination of the widely used fifth-order WENO spatial discretization (WENO5) and the forward Euler time integration method is linearly unstable when numerically integrating hyperbolic conservation laws. Consequently it is not convergent. Furthermore we show that all two-stage, second-order explicit Runge–Kutta (ERK) methods are linearly unstable (and hence do not converge) when coupled with WENO5. We also show that all optimal first- and second-order strong-stability-preserving (SSP) ERK methods are linearly unstable when coupled with WENO5. Moreover the popular three-stage, third-order SSP(3,3) ERK method offers no linear stability advantage over non-SSP ERK methods, including ones with negative coefficients, when coupled with WENO5. We give new linear stability criteria for combinations of WENO5 with general ERK methods of any order. We find that a sufficient condition for the combination of an ERK method and WENO5 to be linearly stable is that the linear stability region of the ERK method should include the part of the imaginary axis of the form $[-\iota \mu,\iota \mu]$ for some $\mu>0$. The linear stability analysis also provides insight into the behavior of ERK methods applied to nonlinear problems and problems with discontinuous solutions. We confirm the assertions of our analysis by means of numerical tests.

On the use of boundary-integral equation methods for unsteady free-surface flows

The Mixed Eulerian-Lagrangian Methods (MEL) forfree-surface potential flows solved by boundary-integral equations (BIEs) is considered, and the diffusion and dispersion errors are studied in the discrete linearized problem. The diffusion error is the base for the stability analysis of the scheme; both the errors give indications on the accuracy of the numerical solution. The study is divided into two steps: comparison of the discrete dispersion relation with the analytical solution and coupling with different time-integration schemes. In particular, a stability analysis of the Runge-Kutta and Taylor-expansion schemes, previously not given in the literature, is addressed. It is shown that MEL methods based on first- and second-order explicit Runge-Kutta and Taylor-expansion schemes are unstable, regardless of the technique adopted to discretize the BIEs. Higher-order Runge-Kutta and Taylor-expansion schemes lead to conditionally stable methods. Known results for explicit, implicit and explicit-implicit Euler schemes are recovered by the present analysis. The theoretical predictions of the errors are confirmed for two different boundary-element techniques: a high-order panel method based on B-Splines to solve for the velocity potential and a spectrally-accurate method based on the Euler-McLaurin summation formula to solve directly for the velocity field.

Numerical Study on Reduction of Transonic Blade-Vortex Interaction Noise

Reduction of noise caused by transonic blade-vortex interaction (BVI) is investigated numerically. The near and midfield flowfields are obtained by an Euler solver. The Euler solver is based on a third-order upwind finite volume scheme in space and a second-order explicit Runge-Kutta scheme in time. Far-field noise is then obtained from the Kirchhoff method. Two control techniques, blowing/suction and porous wall on the airfoil surface, are investigated to reduce two dominant disturbances, transonic and compressibility waves, in the BVI noise signature. Numerical results indicate that the blowing/suction control technique reduces the fluctuations generated by the transonic wave but has little influence on the compressibility wave unless it is employed at the leading edge of the blade. With surface porosity control both compressibility and transonic waves are satisfactorily reduced.

Numerical study of transonic blade-vortex interaction

Numerical investigations of aerodynamic sound generation due to transonic blade-vortex interaction were performed numerically. The numerical method is based on a third-order upwind finite-volume scheme in space and a second-order explicit Runge-Kutta scheme in time. A standard transonic blade-vortex interaction is presented to demonstrate two dominant sound waves, transonic and compressibility waves. The fluctuation dilatation is presented to identify these two significant sound waves which travel upstream. Three unsteady shock wave motions, type A, B and C identified by Tijdeman and Seebass, were simulated by changing the physical parameters, such as Mach number, vortex strength, and initial position of the vortex.