Pao-Hsiung Chiu

A differentially interpolated direct forcing immersed boundary method for predicting incompressible Navier-Stokes equations in time-varying complex geometries

A dispersion-relation-preserving dual-compact scheme developed in Cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time-varying domains. The artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body. We develop in this study a differential-based interpolation scheme which can be easily extended to three-dimensional simulation. The results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems. The accuracy and fidelity of the IB flow solver developed to predict flows with irregular boundaries are therefore demonstrated.

On the development of a dispersion-relation-preserving dual-compact upwind scheme for convection-diffusion equation

In this paper a dual-compact scheme, which accommodates a better dispersion relation for the convective terms shown in the transport equation, is proposed to enhance the convective stability of the convection-diffusion equation by virtue of the increased dispersive accuracy. The dispersion-relation-preserving compact scheme has been rigorously developed within the three-stencil point framework through the dispersion and dissipation analyses. To verify the proposed method, several problems that are amenable to the exact and benchmark solutions will be investigated. The results with good rates of convergence are demonstrated for all the investigated problems.

A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave equation

The paper presents an iterative algorithm for studying a nonlinear shallow-water wave equation. The equation is written as an evolution equation, involving only first-order spatial derivatives, coupled with the Helmholtz equation. We propose a two-step iterative method that first solves the evolution equation by the implicit midpoint rule and then solves the Helmholtz equation using a three-point sixth-order compact scheme. The first-order derivative terms in the first step are approximated by a sixth-order dispersion-relation-preserving scheme that preserves the physically inherent dispersive nature. The compact Helmholtz solver, on the other hand, allows us to use relatively few nodal points in a stencil, while achieving a higher-order accuracy. The midpoint rule is a symplectic time integrator for Hamiltonian systems, which may be a preferable method to solve the spatially discretized evolution equation. To give an assessment of the dispersion-preserving scheme, we provide a detailed analysis of the dispersive and dissipative errors of this algorithm. Via a variety of examples, we illustrate the efficiency and accuracy of the proposed scheme by examining the errors in different norms and providing the rates of convergence of the method. In addition, we provide several examples to demonstrate that the conserved quantities of the equation are well preserved by the implicit midpoint time integrator. Finally, we compare the accuracy, elapsed computing time, and spatial and temporal rates of convergence among the proposed method, a complete integrable particle method, and the local discontinuous Galerkin method.

Development of a dispersion relation-preserving upwinding scheme for incompressible Navier-Stokes equations on nonstaggered grids

ABSTRACT In this article a scheme which preserves the dispersion relation for convective terms is proposed for solving the two-dimensional incompressible Navier–Stokes equations on nonstaggered grids. For the sake of computational efficiency, the splitting methods of Adams-Bashforth and Adams-Moulton are employed in the predictor and corrector steps, respectively, to render second-order temporal accuracy. For the sake of convective stability and dispersive accuracy, the linearized convective terms present in the predictor and corrector steps at different time steps are approximated by a dispersion relation-preserving (DRP) scheme. The DRP upwinding scheme developed within the 13-point stencil framework is rigorously studied by virtue of dispersion and Fourier stability analyses. To validate the proposed method, we investigate several problems that are amenable to exact solutions. Results with good rates of convergence are obtained for both scalar and Navier–Stokes problems.

An effective explicit pressure gradient scheme implemented in the two-level non-staggered grids for incompressible Navier-Stokes equations

In this paper, an improved two-level method is presented for effectively solving the incompressible Navier-Stokes equations. This proposed method solves a smaller system of nonlinear Navier-Stokes equations on the coarse mesh and needs to solve the Oseen-type linearized equations of motion only once on the fine mesh level. Within the proposed two-level framework, a prolongation operator, which is required to linearize the convective terms at the fine mesh level using the convergent Navier-Stokes solutions computed at the coarse mesh level, is rigorously derived to increase the prediction accuracy. This indispensable prolongation operator can properly communicate the flow velocities between the two mesh levels because it is locally analytic. Solution convergence can therefore be accelerated. For the sake of numerical accuracy, momentum equations are discretized by employing the general solution for the two-dimensional convection-diffusion-reaction model equation. The convective instability problem can be simultaneously eliminated thanks to the proper treatment of convective terms. The converged solution is, thus, very high in accuracy as well as in yielding a quadratic spatial rate of convergence. For the sake of programming simplicity and computational efficiency, pressure gradient terms are rigorously discretized within the explicit framework in the non-staggered grid system. The proposed analytical prolongation operator for the mapping of solutions from the coarse to fine meshes and the explicit pressure gradient discretization scheme, which accommodates the dispersion-relation-preserving property, have been both rigorously justified from the predicted Navier-Stokes solutions.

Development of an Upwinding Scheme through the Minimization of Modified Wavenumber Error for the Incompressible Navier-Stokes Equations

In this article we develop a computationally stable and dispersively accurate convective scheme for the incompressible Navier-Stokes equations predicted in non-staggered grids. To enhance the convective stability and improve the dispersion accuracy, the convective terms are approximated by conducting the dispersion analysis to minimize dispersion relation error and Fourier stability analysis. To validate the proposed third-order-accurate two-dimensional numerical scheme, we solve four problems that are all amenable to exact solutions and the lid-driven cavity problem investigated at high Reynolds numbers. Results with good rates of convergence are obtained for the scalar and Navier-Stokes problems.

On the development of a high-order compact scheme for exhibiting the switching and dissipative solution natures in the Camassa-Holm equation

In this paper a two-step iterative solution algorithm for solving the Camassa-Holm equation, which involves only the first-order derivative term, is presented. In each set of the u-P and u-m differential equations, one is governed by the inviscid nonlinear convection-reaction equation for the time-evolving fluid velocity component along the horizontal direction. The other equation is known as the inhomogeneous Helmholtz equation. The resulting reduction of differential order facilitates us to develop the flux discretization scheme in a stencil with comparatively fewer points. For accurately predicting the unidirectional propagation of the shallow water wave, the modified equation analysis for eliminating several leading discretization error terms and the Fourier analysis for minimizing a particular type of wave-like error are employed. In this study, the fifth-order spatially accurate combined compact upwind scheme is developed in a three-point stencil for approximating the first-order derivative term. For the purpose of retaining a long-term accurate Hamiltonian and multi-symplectic geometric structures in Camassa-Holm equation, the time integrator (or time-stepping scheme) chosen in this study should conserve symplecticity. Another main emphasis of conducting the present calculation of Camassa-Holm equation is to shed light on the conservation of Hamiltonians up to the time before wave breaking. We also intended to elucidate the switching scenario by virtue of the peakon-peakon interaction problem and the dissipative scenario after the time of head-on collision in the peakon-antipeakon interaction problem.

A sixth-order dual preserving algorithm for the Camassa-Holm equation

The paper presents a sixth-order numerical algorithm for studying the completely integrable Camassa-Holm (CH) equation. The proposed sixth-order accurate method preserves both the dispersion relation and the Hamiltonians of the CH equation. The CH equation in this study is written as an evolution equation, involving only the first-order spatial derivatives, coupled with the Helmholtz equation. We propose a two-step method that first solves the evolution equation by a sixth-order symplectic Runge-Kutta method and then solves the Helmholtz equation using a three-point sixth-order compact scheme. The first-order derivative terms in the first step are approximated by a sixth-order dispersion-relation-preserving scheme that preserves the physically inherent dispersive nature. The compact Helmholtz solver, on the other hand, allows us to use relatively few nodal points in a stencil, while achieving a higher-order accuracy. The sixth-order symplectic Runge-Kutta time integrator is preferable for an equation that possesses a Hamiltonian structure. We illustrate the ability of the proposed scheme by examining examples involving peakon or peakon-like solutions. We compare the computed solutions with exact solutions or asymptotic predictions. We also demonstrate the ability of the symplectic time integrator to preserve the Hamiltonians. Finally, via a smooth travelling wave problem, we compare the accuracy, elapsed computing time, and rate of convergence among the proposed method, a second-order two-step algorithm, and a completely integrable particle method.

Application of a DRP Upwinding Scheme in Immersed Boundary Method

In this technical note we will presnet the idea of the mimic-quadric-interpolation immersed boundary (MQI-IB) method. A Dispersion-Relation-Preserving (DRP) upwinding scheme for the convective terms developed in Cartesian grids is also applied within the present analysis framework.