Fig 1 - uploaded by Tung-Lin Tsai
Content may be subject to copyright.
Schematic grid diagram for cubic-spine interpolation
Source publication
The characteristics method by using the cubic-spline interpolation is comparable to the Holly-Preissmann scheme in solving the advection portion of the advection-diffusion equation. In order to conduct a cubic-spline interpolation, an additional constraint must be specified at each endpoint. In general, four types of endpoint constraints are availa...
Contexts in source publication
Context 1
... x and t represent the grid size and time step, respec- tively see Fig. 1. Cr is the Courant number. The schematic dia- gram of the characteristic trajectory is shown in Fig. 1. h is the unknown concentration of grid point h at time level n, which is to be solved. f is the concentration of grid point f at time level n 1, in which concentrations of all grid points are known. Since, in general, the foot of the ...
Context 2
... x and t represent the grid size and time step, respec- tively see Fig. 1. Cr is the Courant number. The schematic dia- gram of the characteristic trajectory is shown in Fig. 1. h is the unknown concentration of grid point h at time level n, which is to be solved. f is the concentration of grid point f at time level n 1, in which concentrations of all grid points are known. Since, in general, the foot of the trajectory, x f , does not coincide with grid points, one must employ some form of interpolation to ...
Context 3
... may develop a cubic-spline interpolation function for evaluating h corresponding to all the known concentrations at time level n1, that is, i n1 , i0,1,...,M shown in Fig. 1. In the cubic-spline interpolation, the second derivative is a continuous piecewise linear function and can be expressed ...
Similar publications
A kind of spectral meshless radial point interpolation method is proposed to degenerate parabolic equations arising from the spatial diffusion of biological populations and satisfactory agreements is archived. This method is based on collocation methods with mesh-free techniques as a background. In contrast to the finite-element method and those me...
In this study, effects of operator splitting methods to the solution of advection-diffusion equation are examined. Within the context of this work two operator splitting methods, Lie-Trotter and Strang splitting methods were used and comparisons were made through various Courant numbers. These methods have been implemented to advection-diffusion eq...
The four-equation model consists of two equations for the fluid and the other two for the pipe, describing the axial vibration of liquid-pipes. Fluid-structure interaction (FSI) is significantly considered in this model, typically solved with the method of characteristics (MOC) which introduces the numerical error inevitably. When friction and damp...
Recently we proposed a linear surface scheme for the Method Of Characteristics (MOC) on unstructured meshes. This scheme provided a linear spatial expansion for the flux in arbitrary geometries, based on surface values interpolation. In this paper we present a further development of this method, accepting now the same kind of spatial variation insi...
The CIP-MOC (Constrained Interpolation Profile/Method of Characteristics) is proposed to solve the tide wave equations with large time step size. The bottom topography and bottom friction, which are very important factor for the tidal wave model, are included to the equation of Riemann invariants as the source term. Numerical experiments demonstrat...
Citations
... Many studies have proposed characteristic finite difference (CFD) methods to overcome the numerical dissipations and non-physical oscillations which arise in the convection-dominated problems [36], [39][40][41][42][43][44]. In the CFD methods, traditional finite difference is used for diffusion terms and the convection terms are treated by the method of characteristics. ...
... Obviously, this complicates the calculations and causes errors. Therefore, in many studies the time step size is restricted to ensure that the characteristics lines do not go through the boundaries [41], [45], [46]. ...
In this study, a class of GPU-based corrected explicit-implicit domain decomposition schemes (GCEIDD) is proposed to accelerate the solution of convection-dominated diffusion problems on GPUs. All of these methods take advantage of the fractional steps and corrected explicit-implicit domain decomposition techniques. In each sweep, the domain is decomposed into many strip-divided subdomains. The aim is to reduce the size and increase the number of tri-diagonal systems to provide enough parallelism to keep the GPU occupied. In different steps of the algorithms, we use low-complexity schemes with a high degree of parallelism. The generic form of GCEIDD allows different schemes for the prediction, correction, and one-dimensional implicit solution. We have proposed two methods based on modified upwind and characteristics finite difference and a combined method that bypasses the limitations of these two. Also, important aspects of CUDA programming, which are critical to the implementation of the GCEIDD algorithm, are discussed in detail. For solving tri-diagonal systems, a three-stage strategy is proposed, which creates a balance between occupancy, cache hit rate, and shared memory usage and reduces cache contentions. Results show that GCEIDD has good accuracy and stability even when the number of subdomains is very large. The proposed method can accelerate the solution by a factor of up to 3 compared with the GPU implementation of the classic fractional step methods.
... A variety of numerical methods have been proposed for solving ADE, such as the method of characteristics [3][4][5][6][7][8][9], the finite difference method [10][11][12][13][14][15][16][17][18][19], the finite element method [20][21][22][23], the differential quadrature method [24,25], the Lattice Boltzman method [26], and the meshless method [27][28][29][30]. In these studies, solutions of the ADE with constant parameters have been obtained. ...
A high-accuracy numerical method based on a sixth-order combined compact difference scheme and the method of lines approach is proposed for the advection-diffusion transport equation with variable parameters. In this approach, the partial differential equation representing the advection-diffusion equation is converted into many ordinary differential equations. These time-dependent ordinary differential equations are then solved using an explicit fourth order Runge-Kutta method. Three test problems are studied to demonstrate the accuracy of the present methods. Numerical solutions obtained by the proposed method are compared with the analytical solutions and the available numerical solutions given in the literature. In addition to requiring less CPU time, the proposed method produces more accurate and more stable results than the numerical methods given in the literature.
... For this reason, many researchers developed different methods to solve Eq. (2) accurately. Some of these methods are classical finite difference method [2], high-order finite element method [3], high-order finite difference methods [4,5], green element method [6], cubic and extended B-spline collocation methods [7][8][9], cubic, quartic and quintic B-spline differential quadrature methods [10,11], method of characteristics unified with splines [12][13][14], cubic trigonometric B-spline approach [15], Taylor collocation and Taylor-Galerkin methods [16], Lattice Boltzmann method [17]. Moreover, non-linear advection-diffusion equation is studied in [18]. ...
This paper describes a numerical solution for the advection-diffusion equation. The proposed method is based on the operator splitting method which helps to obtain accurate solutions. That is, instead of sum, the operators are considered separately for the physical compatibility. In the process, method of characteristics combined with cubic spline interpolation and Saulyev method are used in sub-operators, respectively. After guaranteeing the convergence of the method the efficiency is also tested on one-dimensional advection-diffusion problem for a wide range of Courant numbers which plays a crucial role on the convergence of the solution. The obtained results are compared with the analytical solution of the problem and other solutions which are available in the literature. It is revealed that the proposed method produces good approach not only for small Caurant numbers but also big ones even though it is explicit method.
... For this reason, many researchers developed different methods to solve Eq. (2) accurately. Some of these methods are classical finite difference method [2], high-order finite element method [3], high-order finite difference methods [4,5], green element method [6], cubic and extended B-spline collocation methods [7][8][9], cubic, quartic and quintic B-spline differential quadrature methods [10,11], method of characteristics unified with splines [12][13][14], cubic trigonometric B-spline approach [15], Taylor collocation and Taylor-Galerkin methods [16], Lattice Boltzmann method [17]. Moreover, non-linear advection-diffusion equation is studied in [18]. ...
This paper describes a numerical solution for the advection-diffusion equation. The proposed method is based on the operator splitting method which helps to obtain accurate solutions. That is, instead of sum, the operators are considered separately for the physical compatibility. In the process, method of characteristics combined with cubic spline interpolation and Saulyev method are used in sub-operators, respectively. After guaranteeing the convergence of the method the efficiency is also tested on one-dimensional advection-diffusion problem for a wide range of Courant numbers which plays a crucial role on the convergence of the solution. The obtained results are compared with the analytical solution of the problem and other solutions which are available in the literature. It is revealed that the proposed method produces good approach not only for small Caurant numbers but also big ones even though it is explicit method.
... However, since this equation contains two different physical processes such as advection and diffusion, the precise numerical solution is quite difficult. To overcome this difficulty such as classical finite difference method [4], high-order finite element method [5], high-order finite difference methods [6,7], green element method [8], cubic and extended Bspline collocation methods [9][10][11], cubic, quartic and quintic B-spline differential quadrature methods [12,13], method of characteristics unified with splines [14][15][16], cubic trigonometric B-spline approach [17] Taylor collocation and Taylor-Galerkin methods [18] , Lattice Boltzmann method [19] have been developed. In addition, with the help of operator splitting methods, the appropriate methods for the physical processes of the problem can be combined. ...
This study proposes a semi-Lagrangian scheme for numerical simulation of advection-diffusion equation. The proposed method provides unconditional stability and highly accurate solutions even at large time steps. Another advantage of this method is that it requires a low computational time. Accuracy of the method is tested by a numerical application.
... The basis of the proposed method is based on the backward time-line characteristics approach. Tsai et al. [7] investigated effects of the endpoint constraints which are used in the characteristics method with cubic spline interpolation, on the solution of advection process. Verma et al. [8], with the help of Lie-Trotter operator splitting method, used the MacCormack scheme and the Crank-Nicolson finite difference scheme for the solution of the advection and diffusion processes, respectively. ...
In this study, effects of operator splitting methods to the solution of advection-diffusion equation are examined. Within the context of this work two operator splitting methods, Lie-Trotter and Strang splitting methods were used and comparisons were made through various Courant numbers. These methods have been implemented to advection-diffusion equation in one-dimension. Numerical solutions of advection and dispersion processes were carried out by a characteristics method with cubic spline interpolation (MOC-CS) and Crank-Nicolson (CN) finite difference scheme, respectively. Obtained results were compared with analytical solutions of the problems and available methods in the literature. It is seen that MOC-CS-CN method has lower error norm values than the other methods. MOC-CS-CN produces accurate results even while the time steps are great.
... In Driessen and Dohner [2001], finite elements and boundary elements are combined in order to solve advection-diffusion problems with variable advection in an infinite domain. There exist alternative methods for numerical solution of advection-diffusion type equations including operator splitting e.g., Khan and Liu [1998] and Remesikova [2007], spline-type techniques e.g., Funaro and Pontrelli [1999], Tsai et al. [2004], Zoppou et al. [2000] and Irk et al. [2015] and spectral methods e.g., Sakai and Kimura [2005], among many others [Wang et al., 1999;Witeka et al., 2008;Isshiki, 2014;Frutos et al., 2014;Ravnik and Skerget, 2014]. ...
Generalized integral representation method (GIRM) is designed to numerically solve initial and boundary value problems for differential equations. In this work, we develop numerical schemes based on 1- and 2-step GIRMs for evaluation of the two-dimensional problem of advective diffusion in an infinite domain. Accurate approximate solutions are obtained in both cases of GIRM and compared to the exact ones. The derivation of GIRM is straightforward and implementation is simple.
... Pepper et al. (1979) and Okamoto et al. (1998) have solved the one-dimensional advection equation by using a spline interpolation technique called a quasi-Lagrangian cubic spline method. The characteristic methods integrated with splines have been proposed to solve advection-diffusion equation (Szymkiewicz, 1993;Tsai et al., 2004;. Gardner and Dag (1994) have set up the cubic B-spline Galerkin method to solve the advection-diffusion equation. ...
... Pepper et al. (1979) and Okamoto et al. (1998) have solved the one-dimensional advection equation by using a spline interpolation technique called a quasi-Lagrangian cubic spline method. The characteristic methods integrated with splines have been proposed to solve advection-diffusion equation (Szymkiewicz, 1993;Tsai et al., 2004;. Gardner and Dag (1994) have set up the cubic B-spline Galerkin method to solve the advection-diffusion equation. ...
The collocation method based on the extended B-spline functions as trial functions is set up to find numerical solutions of the advection-diffusion equation numerically. The transport of the pollution is simulated by way of using solutions of the advection-diffusion equation. Comparative results of use of both B-spline and extended B-spline in the collocation method are illustrated. It is concluded that a suitable selection of the free parameter of extended B-spline is shown to provide less error for the numerical solutions of the advection-diffusion equation with some boundary and initial conditions.
... Remarkable research studies have been conducted in order to solve advection-dispersion equation numerically like method of characteristic with Galerkin method [2], finite difference method [3][4][5], high-order finite element techniques [6], high-order finite difference methods [7][8][9][10][11][12][13][14][15][16][17][18][19][20], green element method [21], cubic B-spline [22], cubic Bspline differential quadrature method [23], method of characteristics integrated with splines [24][25][26], Galerkin method with cubic B-splines [27], Taylor collocation and Taylor-Galerkin methods [28], B-spline finite element method [29], least squares finite element method (FEMLSF and FEMQSF) [30], lattice Boltzman method [31], Taylor-Galerkin B-spline finite element method [32], and meshless method [33,34]. ...
In this paper, numerical simulation of advective-diffusive contaminant transport are carried out by using high-order compact finite difference schemes combined with second-order MacCormack and fourth-order Runge-Kutta time integration schemes. Both of two schemes have accuracy of sixth-order in space. A sixth-order MacCormack scheme is proposed first time within this study. For the aim of demonstrating efficiency and high-order accuracy of the current methods, numerical experiments have been done. The schemes are implemented to solve two test problems with known exact solutions. It has been exhibited that the methods are capable of succeeding high accuracy and efficiency with minimal computational effort, by comparisons of the computed results with exact solutions.
