Applying multiquadric quasi-interpolation to solve Burgers' equation

In this paper, we develop a kind of univariate multiquadric (MQ) quasi-interpolation and use it to solve Burgers' equation (with viscosity). At first we construct the MQ quasi-interpolation, which possesses the properties of linear reproducing and preserving monotonicity. Next we obtain the numerical scheme, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable. Then, we verify our method for two examples with distinguishing initial value condition. One example is tested for three Reynolds number, that is, R = 10, R = 100, and R = 10,000. From the numerical experiments, we see that the presented method in this paper is valid although the accuracy of the technique is not higher than Hon and Mao's one. Another example is used to examine the travelling of the shock. We can improve the accuracy by selecting an appropriate shape parameter and using a higher accurate MQ quasi-interpolation. The advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement. (c) 2005 Elsevier Inc. All rights reserved.