# PERFORMANCE OF RICHARDSON EXTRAPOLATION ON SOME NUMERICAL METHODS FOR A SINGULARLY PERTURBED TURNING POINT PROBLEM WHOSE SOLUTION HAS BOUNDARY LAYERS

• Munyakazi, Justin B. (Department of Mathematics and Applied Mathematics University of the Western Cape) ;
• Patidar, Kailash C. (Department of Mathematics and Applied Mathematics University of the Western Cape)
• Accepted : 2014.02.03
• Published : 2014.07.01
• 74 14

#### Abstract

Investigation of the numerical solution of singularly perturbed turning point problems dates back to late 1970s. However, due to the presence of layers, not many high order schemes could be developed to solve such problems. On the other hand, one could think of applying the convergence acceleration technique to improve the performance of existing numerical methods. However, that itself posed some challenges. To this end, we design and analyze a novel fitted operator finite difference method (FOFDM) to solve this type of problems. Then we develop a fitted mesh finite difference method (FMFDM). Our detailed convergence analysis shows that this FMFDM is robust with respect to the singular perturbation parameter. Then we investigate the effect of Richardson extrapolation on both of these methods. We observe that, the accuracy is improved in both cases whereas the rate of convergence depends on the particular scheme being used.

#### Keywords

singular perturbations;turning point problems;boundary layers;fitted operator finite difference methods;fitted mesh finite difference method;Richardson extrapolation;error estimates

#### Acknowledgement

Supported by : South African National Research Foundation

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