Основні поняття
The paper constructs and proves the convergence of a finite volume scheme for scalar conservation laws with compactly heterogeneous flux functions that do not satisfy the classical Kruzhkov framework.
Анотація
The paper focuses on the scalar conservation law ∂tu + ∂x(H(x, u)) = 0 with initial condition u0 ∈ L∞(R, R), where the flux function H satisfies the following assumptions:
- Smoothness: H ∈ C3(R2, R)
- Compact Heterogeneity: There exists X > 0 such that for |x| ≥ X, ∂uH(x, u) = 0
- Strong Convexity: For all x ∈ R, u ↦→ ∂uH(x, u) is an increasing C1-diffeomorphism of R onto itself
The main contributions are:
- The authors build a finite volume scheme for this class of conservation laws by treating each interface as a discontinuous flux problem. This allows them to obtain L∞ bounds on the approximate solutions.
- They prove the convergence of the scheme to the unique entropy solution of the conservation law using the compensated compactness method, taking advantage of the genuine nonlinearity of H under the strong convexity assumption.
- As a byproduct, the authors provide an alternate existence result for the Cauchy problem that does not rely on the vanishing viscosity method.
The key steps are:
- Introduce the theory of discontinuous flux and define the notion of entropy solution for conservation laws with discontinuous flux.
- Construct a finite volume scheme that treats the space dependency of the flux function by considering each interface as a discontinuous flux problem.
- Establish the stability of the scheme and derive the necessary a priori estimates using the theory of discontinuous flux.
- Prove the strong compactness of the approximate solutions and pass to the limit to obtain the convergence to the unique entropy solution.