Abstract. We obtain new L 1 contraction results for bounded entropy solutions of Cauchy problems for degenerate parabolic equations. The equations we consider have possibly strongly degenerate local or non-local diffusion terms. As opposed to previous results, our results apply without any integrability assumption on the solutions. They take the form of partial Duhamel formulas and can be seen as quantitative extensions of finite speed of propagation local L 1 contraction results for scalar conservation laws. A key ingredient in the proofs is a new and non-trivial construction of a subsolution of a fully non-linear (dual) equation. Consequences of our results are maximum and comparison principles, new a priori estimates, and in the non-local case, new existence and uniqueness results.

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu − L[ϕ(u)] = f (x, t) in R N × (0, T), where L is a general symmetric Lévy type diffusion operator. Included are both local and nonlocal problems with e.g. L = ∆ or L = −(−∆) α 2 , α ∈ (0, 2), and porous medium, fast diffusion, and Stefan type nonlinearities ϕ. By robust methods we mean that they converge even for nonsmooth solutions and under very weak assumptions on the data. We show that they are L p-stable for p ∈ [1, ∞], compact, and convergent in C([0, T ]; L p loc (R N)) for p ∈ [1, ∞). The first part of this project is given in [36] and contains the unified and easy to use theoretical framework. This paper is devoted to schemes and testing. We study many different problems and many different concrete discretizations, proving that the results of [36] apply and testing the schemes numerically. Our examples include fractional diffusions of different orders, and Stefan problems, porous medium, and fast diffusion nonlinearities. Most of the convergence results and many schemes are completely new for nonlocal versions of the equation, including results on high order methods, the powers of the discrete Laplacian method, and discretizations of fast diffusions. Some of the results and schemes are new even for linear and local problems.

No abstract

No abstract

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu − L σ,µ [ϕ(u)] = f (x, t) in R N × (0, T), where L σ,µ is a general symmetric diffusion operator of Lévy type and ϕ is merely continuous and non-decreasing. We then use this theory to prove convergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion operators L σ,µ are the (fractional) Laplacians ∆ and −(−∆) α 2 for α ∈ (0, 2), discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal Lévy operators, allows us to give a unified and compact nonlocal theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and convergence of the methods under minimal assumptions-including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in [28]. We also present some numerical tests, but extensive testing is deferred to the companion paper [31] along with a more detailed discussion of the numerical methods included in our theory.