The values of in Eq. will depend on the problem coefficients and solution. They will c also depend on In general, they will change monotonically as is decreased, h. h approaching constant values as approaches zero. As a result, the error in any solution h value will, as approaches zero, always reduce when is reduced in a manner that is h h consistent with the order of the scheme. For example, with a second-order accurate scheme, the error in any solution value will reduce by a factor of approximately four when the node spacing is halved. For some types of problems, numerical schemes may not be convergent”, “consistently i.e. the error at a point in the solution may not always decrease as is decreased, even as h approaches zero, or may decrease as is decreased, but not in a way that is consistent h h with the order of the method. The value of for a given scheme can be estimated empirically in a number of ways, p giving an estimated order of convergence (EOC). If an exact solution is known then it can be estimated as [80] where is the error in the solution obtained using a step size [81]. When an exact eh h solution is not available, the EOC can be estimated using where , and are corresponding solution values estimated using step sizes Vh Vh/2 Vh/4 of and respectively [80]. h, h/2 h/4, If the results used to estimate the order of convergence are obtained using a scheme that is consistently convergent for the problem tested, then, as approaches zero, and h assuming that round-off errors in the results are insignificant, the estimated order of convergence should approach the order of accuracy of the scheme (e.g. for a scheme, the value of should approach two as is decreased). EOC h Benchmark numerical solutions A benchmark numerical solution is simply a highly accurate numerical approximation that can be used to estimate errors in less accurate estimates, and to compare the
The values of c in Eq. will depend on the problem coefficients and solution. They will also depend on h. In general, they will change monotonically as h is decreased, approaching constant values as h approaches zero. As a result, the error in any solution value will, as h approaches zero, always reduce when h is reduced in a manner that is consistent with the order of the scheme. For example, with a second-order accurate scheme, the error in any solution value will reduce by a factor of approximately four when the node spacing is halved. For some types of problems, numerical schemes may not be “consistently convergent”, i.e. the error at a point in the solution may not always decrease as h is decreased, even as h approaches zero, or may decrease as h is decreased, but not in a way that is consistent with the order of the method. The value of p for a given scheme can be estimated empirically in a number of ways, giving an estimated order of convergence (EOC). If an exact solution is known then it can be estimated as [80] where eh is the error in the solution obtained using a step size h [81]. When an exact solution is not available, the EOC can be estimated using where Vh , Vh/2 and Vh/4 are corresponding solution values estimated using step sizes of h, h/2 and h/4, respectively [80]. If the results used to estimate the order of convergence are obtained using a scheme that is consistently convergent for the problem tested, then, as h approaches zero, and assuming that round-off errors in the results are insignificant, the estimated order of convergence should approach the order of accuracy of the scheme (e.g. for a scheme, the value of EOC should approach two as h is decreased). Benchmark numerical solutions A benchmark numerical solution is simply a highly accurate numerical approximation that can be used to estimate errors in less accurate estimates, and to compare the
Numerical_modelling_of_convection-reaction-diffusion_problems_using_electrical_analogues.docx