RK4 Scheme Research Articles

Extrapolated local radial basis function collocation method for shallow water problems

This paper provides a novel meshfree numerical method – the extrapolated local-radial-basis-function collocation method (ELRBFCM) – to solve the well-known shallow-water equations (or de Saint-Venant equations). To promote the accuracy and stability of the local-radial-basis-function collocation method (LRBFCM), the extrapolation method is employed in the ELRBFCM. The numerical experiments include the simulation of two linear and three non-linear problems. For the linear problems, the results of the extrapolation method not only have good agreement with analytical solutions, but also show better efficiency than the Euler and RK4 scheme. It is convinced that by the extrapolation process, the approximating order and stability of time integration is automatically controlled. For the non-linear problems, the solutions by the proposed method capture the physical reaction that also obtained in the benchmark by a complex high-order finite-volume method. As a result, it is concluded that even for very long period, the meshfree ELRBFCM with high-order temporal approximation can provide robust, high accurate and great efficient numerical simulation for the shallow-water problems.

A computational modeling of two dimensional reaction–diffusion Brusselator system arising in chemical processes

In this article, the authors proposed a modified cubic B-spline differential quadrature method (MCB-DQM) to show computational modeling of two-dimensional reaction–diffusion Brusselator system with Neumann boundary conditions arising in chemical processes. The system arises in the mathematical modeling of chemical systems such as in enzymatic reactions, and in plasma and laser physics in multiple coupling between modes. The MCB-DQM reduced the Brusselator system into a system of nonlinear ordinary differential equations. The obtained system of nonlinear ordinary differential equations is then solved by a four-stage RK4 scheme. Accuracy and efficiency of the proposed method successfully tested on four numerical examples and obtained results satisfy the well known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point $$(B,A/B)$$ if $$1-A+B^{2}>0$$ .

A Higher Order Numerical Scheme for Some Nonlinear Differential Equations: Models in Biology

In this paper, we develop a numerical scheme based on differential quadrature method to solve nonlinear generalizations of the Fisher and Burgers’ equations with the zero flux on the boundary. In construction of the numerical scheme, quasilinearization is used to tackle the nonlinearity of the problem, which is followed by semi-discretization for spatial direction using differential quadrature method (DQM). Semi-discretization of the problem leads to a system of first order initial value problem. For total discretization, we discretize the system of first order initial value problem resulting from the space semi-discretization using the RK4 scheme with constant step length. The method is analyzed for stability and convergence. Finally, the efficiency of the method is derived via numerical comparison between their numerical solution and the exact solution. L 2 and L ∞ error norms demonstrate the accuracy of the proposed method.

A 3D immersed interface method for fluid–solid interaction

In immersed interface methods, solids in a fluid are represented by singular forces in the Navier–Stokes equations, and flow jump conditions induced by the singular forces directly enter into numerical schemes. This paper focuses on the implementation of an immersed interface method for simulating fluid–solid interaction in 3D. The method employs the MAC scheme for the spatial discretization, the RK4 scheme for the time integration, and an FFT-based Poisson solver for the pressure Poisson equation. A fluid–solid interface is tracked by Lagrangian markers. Intersections of the interface with MAC grid lines identify finite difference stencils on which jump contributions to finite difference schemes are needed. To find the intersections and to interpolate jump conditions from the Lagrangian markers to the intersections, parametric triangulation of the interface is used. The velocity of the Lagrangian markers is interpolated directly from surrounding MAC grid nodes with interpolation schemes accounting for jump conditions. Numerical examples demonstrate that (1) the method has near second-order accuracy in the infinity norm for velocity, and the accuracy for pressure is between first and second order; (2) the method conserves the volume enclosed by a no-penetration boundary; and (3) the method can efficiently handle multiple moving solids with ease.

Numerical description of acoustic sources in airframe noise

AbstractThe origin of airframe noise in subsonic flows is mainly related to vorticity perturbations which i) interact with geometric inhomogeneities (egdes, steps, slots), ii) are subject to strong convective accelerations of the mean flow, iii) are strongly amplified in magnitude due to hydrodynamic instability of the mean flow. Therefore, apart from an appropriate modeling of the vorticity perturbations (e.g. causal or stochastic), a suitable Computational Aeroacoustics (CAA) method needs to describe accurately the conversion process from hydrodynamic to acoustic perturbations, i.e. the very noise generation. Aeronautics‐related examples of numerically simulated airframe noise phenomena are presented which were obtained using DLR's CAA‐code based on the high order spatial DRP scheme of Tam & Webb ]1[, the RK4 scheme in time and perfectly matched layers at the freefield boundaries Hu ]2[.

A vector‐parallel scheme for Navier‐Stokes computations at multi‐gigaflop performance rates

AbstractA class of vector‐parallel schemes for solution of steady compressible or incompressible viscous flow is developed and performance studies carried out. The algorithms employ an artificial transient treatment that permits rapid integration to a steady state. In the present work a four‐stage explicit Runge‐Kutta scheme employing variable local step size is utilized for the ODE system integration. The RK‐4 scheme is restructured to allow vectorization and enhance concurrency in the calculation for a streamfunction‐vorticity formulation of the flow problem. The parameters of the resulting RK scheme can be selected to accelerate convergence of the RK recursion. Four main procedures are considered which permit vector‐parallel solution: a Jacobi update, a hybrid of the Jacobi and Gauss‐Seidel method, red‐black ordering and domain decomposition. Numerical performance studies are conducted with a representative viscous incompressible flow calculation. Results indicate that a scheme involving domain decomposition with a Gauss‐Seidel type of update for the RK four‐stage scheme is most effective and provides performance in excess of 8 Gflops on the Cray C‐90.