Well balanced scheme

Question

When do we say that a numerical scheme is well balanced? I could not find the precise definition. I read that these are the schemes which preserve steady state. But in which sense (like when do we say that a discretized data is a steady state)? Is Godunov scheme for Burgers' equation well balanced? If not what are the examples of well balanced schemes for Burgers' equation?

Answer 1

This notion was presented by J.M. Greenberg and A.Y. Leroux [1]. It consists in imposing that a numerical method preserves the equilibria of a system of balance laws $$ \partial_t \boldsymbol{q} + \partial_x \boldsymbol{f}(\boldsymbol{q}) = \boldsymbol{r}(\boldsymbol{q}) \, . $$ To illustrate, let us consider the case of Burgers' equation with logistic source term $u_t + u u_x = u(1-u)$ which equilibria are $u^*(x)=0$ and $u^*(x)=1$. One can note that the Lax-Friedrichs method $$ u_i^{n+1} = \frac{u_{i-1}^{n} + u_{i+1}^{n}}{2}\left(1 - \frac{\Delta t}{2\,\Delta x}\left(u_{i+1}^{n} - u_{i-1}^{n}\right) \right) + \Delta t\,u_i^n\left(1-u_i^n\right) $$ with initial data $u_i^0 = u^*(x_i)$ preserves the equilibrium. Similarly for the homogeneous Burgers' equation $u_t + uu_x = 0$, Godunov's method is well-balanced.

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[1] Greenberg J.M., Leroux A.Y. 1996 “A Well-Balanced Scheme for the Numerical Processing of Source Terms in Hyperbolic Equations.” SIAM Journal on Numerical Analysis 33(1), 1–16. JSTOR:2158421 doi:10.1137/0733001