A third-order upwind scheme for the advection–diffusion equation using spreadsheets

Karahan, Halil

doi:10.1016/j.advengsoft.2006.10.006

Cited by 21 publications

(12 citation statements)

References 38 publications

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“…The distribution of the Gaussian pulse at = 5 s is computed using the exact solution and compared with the concentration distribution obtained using the CD6 solution as shown in Figure 3. As can be seen in Table 4, the CD6 results in Example 3 are far more accurate comparison to Crank-Nicolson (CN) scheme [12] and third-order finite difference (FD3) scheme [17]. In Table 5, a comparison of analytical and CD6 solutions is carried out for various values of with ℎ = 0.025, Δ = 0.005, Cr = 0.16, and Pe = 4.…”

Section: Numerical Illustrationsmentioning

confidence: 96%

“…For that reason, several alternative methods are proposed in the literature for solving the ADE with high accuracy [11]. These include method of characteristic with Galerkin method (MOCG) [11], finite difference method [12][13][14], high-order finite element techniques [15], high-order finite difference methods [16][17][18][19][20][21][22][23][24], Green-element method [25], cubic B-spline [26], cubic B-spline differential quadrature method (CBSDQM) [27], method of characteristics integrated with splines (MOCS) [28][29][30], Galerkin method with cubic B-splines (CBSG) [31], Taylor-Collocation (TC) and Taylor-Galerkin (TG) methods [32], B-spline finite element method [33], Least squares finite element method (FEMLSF and FEMQSF) [34], Lattice Boltzmann method [35], Taylor-Galerkin B-spline finite element method [36], and meshless method [37,38].…”

Section: Introductionmentioning

confidence: 99%

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Numerical Solution of Advection-Diffusion Equation Using a Sixth-Order Compact Finite Difference Method

Gürarslan

Karahan

Alkaya

et al. 2013

Mathematical Problems in Engineering

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This study aims to produce numerical solutions of one-dimensional advection-diffusion equation using a sixth-order compact difference scheme in space and a fourth-order Runge-Kutta scheme in time. The suggested scheme here has been seen to be very accurate and a relatively flexible solution approach in solving the contaminant transport equation forPe≤5. For the solution of the present equation, the combined technique has been used instead of conventional solution techniques. The accuracy and validity of the numerical model are verified through the presented results and the literature. The computed results showed that the use of the current method in the simulation is very applicable for the solution of the advection-diffusion equation. The present technique is seen to be a very reliable alternative to existing techniques for these kinds of applications.

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Section: Numerical Illustrationsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Advection-Diffusion Equation Using a Sixth-Order Compact Finite Difference Method

Gürarslan

Karahan

Alkaya

et al. 2013

Mathematical Problems in Engineering

Self Cite

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“…A third-order upwind scheme for the advection-diffusion equation using a simple spreadsheets simulation is proposed in [14]. In [15], a new flux splitting scheme is proposed.…”

Section: Introductionmentioning

confidence: 99%

Numerical Treatment of a Modified MacCormack Scheme in a Nondimensional Form of the Water Quality Models in a Nonuniform Flow Stream

Pochai

2014

Journal of Applied Mathematics

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Two mathematical models are used to simulate water quality in a nonuniform flow stream. The first model is the hydrodynamic model that provides the velocity field and the elevation of water. The second model is the dispersion model that provides the pollutant concentration field. Both models are formulated in one-dimensional equations. The traditional Crank-Nicolson method is also used in the hydrodynamic model. At each step, the flow velocity fields calculated from the first model are the input into the second model as the field data. A modified MacCormack method is subsequently employed in the second model. This paper proposes a simply remarkable alteration to the MacCormack method so as to make it more accurate without any significant loss of computational efficiency. The results obtained indicate that the proposed modified MacCormack scheme does improve the prediction accuracy compared to that of the traditional MacCormack method.

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“…We consider Eq. (1) with the boundary and initial conditions given as follows [11]: g 0 (0, t) = 0.025 √ 0.000625 + 0.02t exp − (0.5 − t)…”

mentioning

confidence: 99%

“…The one-dimensional advection-diffusion of a Gaussian pulse of unit height, centered at x = 1 in a region bounded by 0 ≤ x ≤ 9 with initial conditions, is given as follows [11] (Table I):…”

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confidence: 99%

Taylor‐Galerkin B‐spline finite element method for the one‐dimensional advection‐diffusion equation

Kadalbajoo

Arora

2010

Numerical Methods Partial

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The advection-diffusion equation has a long history as a benchmark for numerical methods. Taylor-Galerkin methods are used together with the type of splines known as B-splines to construct the approximation functions over the finite elements for the solution of time-dependent advection-diffusion problems. If advection dominates over diffusion, the numerical solution is difficult especially if boundary layers are to be resolved. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method with emphasis on treatment of boundary conditions. Taylor-Galerkin methods have been constructed by using both linear and quadratic B-spline shape functions. Results shown by the method are found to be in good agreement with the exact solution.

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A third-order upwind scheme for the advection–diffusion equation using spreadsheets

Cited by 21 publications

References 38 publications

Numerical Solution of Advection-Diffusion Equation Using a Sixth-Order Compact Finite Difference Method

Numerical Solution of Advection-Diffusion Equation Using a Sixth-Order Compact Finite Difference Method

Numerical Treatment of a Modified MacCormack Scheme in a Nondimensional Form of the Water Quality Models in a Nonuniform Flow Stream

Taylor‐Galerkin B‐spline finite element method for the one‐dimensional advection‐diffusion equation

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