Total variation diminishing scheme

Remarks

Example: for $u_t + a u_x = 0$ with $a > 0$, the upwind scheme with $0 \le \nu \le 1$ gives $u_{j+1}^{n+1} - u_j^{n+1} = (1 - \nu)(u_{j+1}^n - u_j^n) + \nu (u_j^n - u_{j-1}^n)$. By the triangle inequality and a shift of the index, $TV(u^{n+1}) \le (1 - \nu) TV(u^n) + \nu\, TV(u^n) = TV(u^n)$. Non-example: the Lax–Wendroff method with $\nu = 1/2$, applied to the step $u_j^n = 1$ for $j \le 0$ and $u_j^n = 0$ for $j \ge 1$, gives $u_0^{n+1} = 1.125$, $u_1^{n+1} = 0.375$ and $u_j^{n+1} = u_j^n$ for the other $j$. So $TV(u^{n+1}) = 0.125 + 0.75 + 0.375 = 1.25 > 1 = TV(u^n)$. Harten's criterion: a scheme that can be written as $u_j^{n+1} = u_j^n + C_{j+1/2} (u_{j+1}^n - u_j^n) - D_{j-1/2} (u_j^n - u_{j-1}^n)$ with $C_{j+1/2} \ge 0$, $D_{j+1/2} \ge 0$ and $C_{j+1/2} + D_{j+1/2} \le 1$ for all $j$ is TVD. Every monotone conservative scheme is TVD, and every TVD scheme is monotonicity preserving.