On the Generation of Exact Solutions using the Method of Nearby Problems

Raju, Anil; Roy, Christopher J.; Hopkins, Matthew M.

doi:10.2514/6.2005-684

Cited by 4 publications

(8 citation statements)

References 4 publications

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“…In our previous work 7 we attempted to achieve good fits to the underlying numerical solution using global polynomials. These global polynomials tended to provide poor approximations of the numerical solution both at the boundaries and in the vicinity of sharp gradients.…”

Section: One-dimensional Fitsmentioning

confidence: 99%

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Discretization error estimation and exact solution generation using the method of nearby problems.

Sinclair

Raju

Kurzen

et al. 2011

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The Method of Nearby Problems (MNP), a form of defect correction, is examined as a method for generating exact solutions to partial differential equations and as a discretization error estimator. For generating exact solutions, four-dimensional spline fitting procedures were developed and implemented into a MATLAB code for generating spline fits on structured domains with arbitrary levels of continuity between spline zones. For discretization error estimation, MNP/defect correction only requires a single additional numerical solution on the same grid (as compared to Richardson extrapolation which requires additional numerical solutions on systematically-refined grids). When used for error estimation, it was found that continuity between spline zones was not required. A number of cases were examined including 1D and 2D Burgers' equation, the 2D compressible Euler equations, and the 2D incompressible Navier-Stokes equations. The discretization error estimation results compared favorably to Richardson extrapolation and had the advantage of only requiring a single grid to be generated.

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Section: One-dimensional Fitsmentioning

confidence: 99%

“…In our previous work 7 we used global Legendre polynomial fits to the underlying numerical solution. An example for a Reynolds number of 16 was shown in Figure 1a.…”

Section: Hermite Spline Fitmentioning

confidence: 99%

Discretization error estimation and exact solution generation using the method of nearby problems.

Sinclair

Raju

Kurzen

et al. 2011

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“…In our previous work 8 we used global fits (Legendre polynomials). An example for Reynolds number of 16 is shown in Fig.…”

Section: Burgers Equationmentioning

confidence: 99%

Evaluation of Discretization Error Estimators Using the Method of Nearby Problems

Raju

Roy

Hopkins

2005

17th AIAA Computational Fluid Dynamics Conference

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The Method of Nearby Problems (MNP) is employed to evaluate various discretization error estimators. Steadystate Burgers equation is chosen as the example case. Fifth-order Hermite splines are used to generate the analytic curve fit, and the development of these splines in 1D is discussed. Results are presented for Burgers equation corresponding to a viscous shock wave for Reynolds numbers of 8, 64, and 512, as well as for a modified version of Burgers equation with a variable viscosity at a nominal Reynolds number of 64. The results obtained using Hermite splines are compared with the results obtained using cubic splines as well as global Legendre polynomials. Whereas the Legendre polynomial fits exhibited large errors at the boundaries, Hermite splines provide good approximations over the entire domain. Furthermore, in contrast to cubic splines, Hermite splines are shown to provide slopecontinuous source terms due to the use of C 3 continuity at spline zone interfaces. The fifth-order Hermite splines are used to generate an exact solution to a problem nearby the original Burgers equation. Various discretization error estimators are evaluated both for the original Burgers equation, the nearby problem, and a version of Burgers equation with a nonlinear viscosity term. The Method of Nearby Problems is also examined as an error estimator.

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“…Recently, our group has developed a method for both estimating the discretization error and for generating exact solutions, called the method of nearby problems (MNP) [11][12][13][14][15]. Although our earlier work referred to MNP as a new approach to estimating discretization error, an astute reviewer of this manuscript pointed out that this approach has in fact been around for three decades and is commonly referred to as the differential form of defect correction (or the differential correction).…”

Section: Introductionmentioning

confidence: 99%

“…A later paper [12] presented rigorous mathematical theory for the method as applied to first-order quasi-linear ODEs. A more recent effort [13] focused on the generation of nearby problems with exact solutions for steady-state Burgers's equation using global polynomial fits. In this work, both Legendre polynomials and standard polynomials (monomials) were shown to be inadequate due to large errors generated near strong gradients and at boundaries.…”

Section: Introductionmentioning

confidence: 99%

Estimation of Discretization Errors Using the Method of Nearby Problems

2007

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The method of nearby problems is developed as an approach for estimating numerical errors due to insufficient mesh resolution. A key aspect of this approach is the generation of accurate, analytic curve fits to an underlying numerical solution. Accurate fits are demonstrated using fifth-order Hermite splines that provide for solution continuity up to the third derivative, which is recommended for second-order differential equations. This approach relies on the generation of a new problem (and corresponding exact solution) that is "nearby" the original problem of interest, and the nearness requirements are discussed. The method of nearby problems is demonstrated as an accurate discretization error estimator for steady-state Burgers's equation for a viscous shock wave at Reynolds numbers of 8 and 64. A key advantage of using the method of nearby problems as an error estimator is that it requires only one additional solution on the same mesh, as opposed to multiple mesh solutions required for extrapolationbased error estimators. Furthermore, the present results suggest that the method of nearby problems can produce better error estimates than other methods in the preasymptotic regime. The method of nearby problems is also shown to provide a useful framework for evaluating other discretization error estimators. This framework is demonstrated by the generation of exact solutions to problems nearby Burgers's equation as well as a form of Burgers's equation with a nonlinear viscosity variation. Nomenclature a i -f i = spline coefficients E = discrete L 2 norm error function f = general function h = global measure of cell or element size L = differential operator L ref = reference length of domain N = number of mesh points n = number of spline points p = order of accuracy r = grid refinement factor (r > 1) S = spline polynomial s = source term t = time coordinate, s u = local solution variable, m=s x = spatial coordinate, m = scaling constant = viscosity, m 2 =s = solution domain Subscripts exact = exact solution to differential equation i = spline zone number j = node number k = mesh level (1 finest mesh) Superscripts 0 = dimensionless variable, first derivative 00= second derivative 000 = third derivative = scaled variablẽ = numerical solution = estimated exact solution to differential equation

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On the Generation of Exact Solutions using the Method of Nearby Problems

Cited by 4 publications

References 4 publications

Discretization error estimation and exact solution generation using the method of nearby problems.

Discretization error estimation and exact solution generation using the method of nearby problems.

Evaluation of Discretization Error Estimators Using the Method of Nearby Problems

Estimation of Discretization Errors Using the Method of Nearby Problems

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