Primitive, Conservative and Adaptive Schemes for Hyperbolic Conservation Laws

For the last two or three decades, it has become an accepted practice to utilise conservative methods when solving numerically hyperbolic conservation laws. Shock waves are the solution features that demand the utilisation of conservative methods. Practical computational experience shows that the use of a non-conservative method results in the wrong shock strength and thus the wrong propagation speed. To some extent an exception is the Random Choice Method (RCM) of Glimm (Glimm, 1965). This method is non-conservative and yet it gives the correct shock strength and, on average the correct propagation speed. Theoretically, Lax and Wendroff proved in 1960 (Lax and Wendroff, 1960) that if a conservative method converges, it does so to a weak solution of the conservation laws. Today it is known, (Harten, 1983), that if the scheme also satisfies an entropy condition, then the converged solution is the physical weak solution. Hence, there are very good reasons for utilising conservative methods.