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Parametrized maximum principle preserving limiter for finite difference WENO schemes solving convection-dominated diffusion equations. (English) Zbl 1290.65077
The main purpose of the paper is to provide a general framework for the design of conservative high-order integration schemes for solving the nonlinear convection-diffusion equation \(u_t + f(u)_x = A(u)_{xx}\), \(u(x,0) = u_0(x)\), with periodic boundary conditions, and its generalization to the 2-dimensional case, when the problem is convection-dominated. More specifically, the goal is to construct arbitrarily high-order finite difference WENO (weighted essentially non-oscillatory) methods able to preserve the discrete analogue of a strict maximum principle the equation possesses when \(A'(u) > 0\), namely that \(u(x,t) \in [ u_m, u_M]\) for \(t>0\), where \(u_M = \max_x u_0(x)\) and \(u_m = \min_x u_0(x)\). The proposed approach involves several steps: (1) a first order monotone scheme preserving the maximum principle is chosen (for instance, the first-order Lax-Friedrichs scheme for the convection part and the second-order central difference scheme for the diffusion part); (2) high-order finite difference Runge-Kutta WENO schemes are designed for the problem in a conservative way, and (3) a class of parametrized flux limiters are derived guaranteeing that the previous schemes satisfy the discrete maximum principle. This is done by replacing the numerical flux by a conveniently modified one involving the low-order monotone flux that satisfies the maximum principle preserving (MPP) property.
The schemes thus constructed preserve the corresponding discrete maximum principle and maintain the designed order of accuracy without imposing additional restrictions on their Courant-Friedrichs-Lewy condition. This is explicitly proven in the third-order case. In addition, they can be implemented in a straightforward way. The paper includes a large collection of examples ranging from simple problems where the exact solution is known to more complicated systems such as the incompressible Navier-Stokes equation in the vorticity stream-function formulation. In all the examples analyzed, the maximum principle is satisfied with the designed order of accuracy when the solution is smooth.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35B50 Maximum principles in context of PDEs
35Q30 Navier-Stokes equations
35Q35 PDEs in connection with fluid mechanics
76S05 Flows in porous media; filtration; seepage
76M20 Finite difference methods applied to problems in fluid mechanics
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