Numerical Investigation of Thin Film Spreading Driven by Surfactant Using Upwind Schemes

Table 1

Comparing 𝐿₂ and 𝐿_∞ norms for Roe-TVD, Roe-Sweby-TVD, and Roe-BDF schemes where we have taken 𝑛=500 and Δ𝑡=2.5×10⁻⁶.

Method	ℎ(𝑥,𝑡)		𝑔(𝑥,𝑡)
	𝐿2	𝐿∞	𝐿2	𝐿∞
Endogenous surfactant
Roe-TVD versus Roe-Sweby-TVD	0.00694248	0.01257900	0.0015539400	0.002015310
Roe-Sweby-TVD versus Roe-BDF	0.00168707	0.00780482	0.0000845757	0.000414136
Roe-TVD versus Roe-BDF	0.00721239	0.01252690	0.0015349300	0.002011100
Exogenous surfactant
Roe-TVD versus Roe-Sweby-TVD	0.00919323	0.01143190	0.0111612	0.03605900
Roe-Sweby-TVD versus Roe-BDF	0.00636653	0.01057280	0.0021225	0.00227495
Roe-TVD versus Roe-BDF	0.00571950	0.00672977	0.0105552	0.03629150