Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics

We develop error-control based time integration algorithms for compressible fluid dynam-ics(CFD)applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime.Focusing on discontinuous spectral element semidis-cretizations,we design new controllers for existing methods and for some new embedded Runge-Kutta pairs.We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice.We compare a wide range of error-control-based methods,along with the common approach in which step size con-trol is based on the Courant-Friedrichs-Lewy(CFL)number.The optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances,while additionally providing control of the temporal error at tighter tolerances.The numerical examples include challenging industrial CFD applications.

Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates

In this paper,we analyze the explicit Runge-Kutta discontinuous Galerkin(RKDG)methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology.The RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta(TVDRK3)time discretization and upwinding numerical fluxes.By using the energy method,under a standard CourantFriedrichs-Lewy(CFL)condition,we obtain L2stability for general solutions and a priori error estimates when the solutions are smooth enough.The theoretical results are proved for piecewise polynomials with any degree k 1.Finally,since the solutions to this system are non-negative,we discuss a positivity-preserving limiter to preserve positivity without compromising accuracy.Numerical results are provided to demonstrate these RKDG methods.

Symmetric-Adjoint and Symplectic-Adjoint Runge-Kutta Methods and Their Applications

Symmetric and symplectic methods are classical notions in the theory of numerical methods for solving ordinary differential equations.They can generate numerical flows that respectively preserve the symmetry and symplecticity of the continuous flows in the phase space.This article is mainly concerned with the symmetric-adjoint and symplectic-adjoint Runge-Kutta methods as well as their applications.It is a continuation and an extension of the study in[14],where the authors introduced the notion of symplectic-adjoint method of a Runge-Kutta method and provided a simple way to construct symplectic partitioned Runge-Kutta methods via the symplectic-adjoint method.In this paper,we provide a more comprehensive and systematic study on the properties of the symmetric-adjoint and symplecticadjoint Runge-Kutta methods.These properties reveal some intrinsic connections among some classical Runge-Kutta methods.Moreover,those properties can be used to significantly simplify the order conditions and hence can be applied to the construction of high-order Runge-Kutta methods.As a specific and illustrating application,we construct a novel class of explicit Runge-Kutta methods of stage 6 and order 5.Finally,with the help of symplectic-adjoint method,we thereby obtain a new simple proof of the nonexistence of explicit Runge-Kutta method with stage 5 and order 5.

Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations

In 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.

Analysis on a Mathematical Model for Tumor Induced Angiogenesis

Tumor-induced angiogenesis is the process by which unmetastasized tumors recruit red blood vessels by way of chemical stimuli to grow towards the tumor for vascularization and metastasis. We model the process of tumor-induced angiogenesis at the tissue level using ordinary and partial differential equations (ODEs and PDEs) that have a source term. The source term is associated with a signal for growth factors from the tumor. We assume that the source term depends on time, and a parameter (time parameter). We use an explicit stabilized Runge-Kutta method to solve the partial differential equation. By introducing a source term into the PDE model, we extend the PDE model used by H. A. Harrington et al. Our results suggest that the time parameter could play some role in understanding angiogenesis.

Extrapolation of GLMs with IRKS Property to Solve the Ordinary Differential Equations

The extrapolation technique has been proved to be very powerful in improving the performance of the approximate methods by large time whether engineering or scientific problems that are handled on computers. In this paper, we investigate the efficiency of extrapolation of explicit general linear methods with Inherent Runge-Kutta stability in solving the non-stiff problems. The numerical experiments are shown for Van der Pol and Brusselator test problems to determine the efficiency of the explicit general linear methods with extrapolation technique. The numerical results showed that method with extrapolation is efficient than method without extrapolation.

Trio-Geometric mean-based three-stage Runge–Kutta algorithm to solve initial value problem arising in autonomous systems

In the present worldwide scenario,plenty of problems arising in science and engineering which can be modeled as differential equations and out of these,autonomous system has become a subject of great interest.Several laws of physics in which time is considered as an independent variable are expressed as autonomous systems.In this paper,Runge–Kutta(RK)three-stage geometric mean method is used to solve the initial value problem arises in autonomous systems.The method is discussed in detail,convergence of method is discussed,the accuracy and efficiency of the method are proved by considering a numerical example.The result is compared to some other methods and proposed method is found to be more efficient.The detailed analysis of error estimation confirms that proposed method is more efficient as compared to other methods.