High–Order and TVD Methods for Scalar Equations

Central to this chapter is the resolution of two contradictory requirements on numerical methods, namely high–order of accuracy and absence of spurious (unphysical) oscillations in the vicinity of large gradients. It is well–known that high–order linear (constant coefficients) schemes produce unphysical oscillations in the vicinity of large gradients. This was illustrated by some numerical results shown in Chap. 5. On the other hand, the class of monotone methods, defined in Sect. 5.2 of Chap. 5, do not produce unphysical oscillations. However, monotone methods are at most first order accurate and are therefore of limited use. These difficulties are embodied in the statement of Godunov’s theorem [216] to be studied in Sect. 13.5.3. One way of resolving the contradiction between linear schemes of high–order of accuracy and absence of spurious oscillations is by constructing non–linear methods.