A note on the shock-capturing properties of some explicit and implicit schemes for solving the 1-D linear advection equation

Abstract

The technique of Minimized Integrated Exponential Error for Low Dispersion and Low Dissipation, (MIEELDLD) was introduced in Appadu and Dauhoo (2011), Appadu and Dauhoo (2009) and extensive work on this technique is reported further in Appadu (In Press), Appadu (2012). The technique enables us to assess the shock-capturing properties of numerical methods. It also allows us to nd suitable values for parameters present in numerical methods in order to optimise their dissipative and dispersive properties (Appadu 2012). This technique basically makes use of a physical quantity called the Integrated Ex- ponential Error for Low Dispersion and Low Dissipation, IEELDLD. In this work, we obtain the IEELDLD for some explicit, quasi-implicit and implicit meth- ods. We use MIEELDLD to obtain an explicit scheme with more effective shock-capturing properties than Gadd and Carpenter's numerical schemes. Also, an implicit method is con- structed which is almost similar to the one derived by Dehghan (2005) and which has also better shock-capturing properties as compared to the Crank-Nicolson method.