ENO-WENO Schemes: Are ENO-WENO schemes non oscillatory for all kinds (linear/non linear) of problems?

Question

Is there an rigorous proof of ENO-WENO schemes being non oscillatory?

Answer 1

Depends on what you mean by 'non-oscillatory'. In their original paper, Harten & Osher (1987), design the ENO schemes with the 'non-oscillatory' property defined as:

In this paper we construct a uniformly second-order approximation, which is nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time.

They also provide proof in the paper (pp. 281). In this sense, and formally, when an ENO scheme is used for interpolations to compute numerical fluxes in solving Hyperbolic Conservation Laws of the form: $$ \frac{\partial u}{\partial t} + \frac{\partial}{\partial x} F(u) = 0, $$ it is guaranteed that the final numerical solution will have equal or less number of extrema when compared to the exact solution.

In practice, this is achieved by using an adaptive stencil that is constructed in a hierarchical fashion to avoid interpolating across non-smooth regions (e.g. large gradients, shocks, etc.). Note that ENO schemes are just one of the ingredients needed for a numerical solution to converge to the viscosity solution of nonlinear PDEs such as Hamilton-Jacobi or Hyperbolic Conservation Laws. Other ingredients include

1. Use of monotone and consistent numerical flux (or Hamiltonian)
2. A TVD integrator in time

For more details please consult the aforementioned paper. I could add more references including those that discuss implementations if needed.

UPDATE:

So why TVD integrators? Note that ENO (or WENO) only quarantines that you do not interpolate across discontinuities. It assumes nothing about what happens to the solution in time. To ensure that the solution remains non-oscillatory, time integration should have the TVD property. TV of a discrete data-set is defined as

$$ TV(u^n) = \sum_{i=0}^N \left|u^n_i - u^n_{i-1}\right| $$

A time integration scheme is said to be TVD if satisfies the following property:

$$ TV(u^{n+1}) \le TV(u^n) $$

Please note that for a time integration scheme, such as RK methods, to be TVD, they need to satisfy certain bounds on the CFL number. This is a bg topic in itself and I suggest you consult the following papers for accurate discussions:

Efficient implementation of essentially non-oscillatory shock-capturing schemes (Shu, C.W. and Oshert, S)

Total-Variation-Diminishing Time Discretizations (C. W. Shu)

Total variation diminishing Runge-Kutta schemes (S. Gottlieb and C. W. Shu)