Convergence to entropy solution for scalar conservation law

Initial value problem:

$$\left. \begin{array}{r} u_t + f(u)_x = 0 \\ u(x,0) = u_o(x) \end{array} \right\}$$ (1)

with $f$ being strictly convex and $u_o \in BV({\mathbb{R}})$.

Viscous form of update scheme:

$$u^{n+1}_i = u^n_i - \frac{\lambda}{2}( f^n_{i+1} - f^n_{i-1}... ...i + 1/2} \Delta u^n_{i+1/2} - Q^n_{i-1/2} \Delta u^n_{i-1/2} )$$ (2)

with $\lambda = \Delta t /\Delta x$ being fixed. The numerical flux function is given by

$$f^n_{i+1/2} = \frac{1}{2}(f^n_i + f^n_{i+1}) - \frac{1}{2\lambda} Q^n_{i+1/2} \Delta u^n_{i+1/2}$$ (3)

Some standard schemes

1. Lax-Friedrichs

   
   

   $$Q^{LF}=1$$
   
   

2. Engquist-Osher

   
   

   $$Q^{EO}_{i+1/2} = \lambda \int^1_0 \vert f'(u_i + \theta (u_{i+1} - u_i)) \vert \textrm{d} \theta$$
   
   

3. Godunov

   
   

   $$Q^{G}_{i+1/2} = \lambda \max_{(s-u_i)(u_{i+1}-s) \ge 0} \frac{ f(u_i) + f(u_{i+1}) - sf(s) }{u_{i+1} - u_i}$$
   
   

4. Murman-Roe

   
   

   $$Q^{MR}_{i+1/2} = \lambda\left\vert \int^1_0 f'(u_i + \theta ... ... f'(u_i) \vert & \mbox{ if } u_i = u_{i+1} \end{array} \right.$$
   
   

$$0 \le Q^{MR} \le Q^G \le Q^{EO} \le Q^{LF} \le 1$$

Tadmor's convergence result:

$$\left. \begin{array}{r} Q^G_{i+1/2} \le Q_{i+1/2} \le 1\\ \lambda \sup\vert f'(s)\vert \le \frac{1}{2} \end{array} \right\}$$ (4)

The restriction of CFL number was increased to 1 by Aiso.

Theorem (Aiso [1]): Fix an arbitrary positive number $\epsilon \le 1$. Suppose that the flux function $f$ is strictly convex. If the numerical viscosity satisfies

$$\max\{ Q^{MR}_{1+1/2}, \epsilon (u_{i+1} - u_i) \} \le Q_{i+1/2} \le 1$$ (5)

then difference approximation (2) converges to the entropy solution under CFL $\le$ 1.

References

1. Aiso H, Admissibility of difference approximation for scalar conservation laws, Hiroshima Math Journal, 23(1):15-61, 1993.

2. Aiso H, A general class of difference approximation for scalar conservation laws converging to the entropy solution and including high resolution ones, 15'th ICNMFD, Monterey, 1996.