The robust finite volume schemes for modeling non-classical surface reactions

A coupled system of nonlinear parabolic PDEs arising in modeling of surface reactions with piecewise continuous kinetic data is studied. The nonclassic conjugation conditions are used at the surface of the discontinuity of the kinetic data. The finite-volume technique and the backward Euler method are used to approximate the given mathematical model. The monotonicity, conservativity, positivity of the approximations are investigated by applying these finite-volume schemes for simplified subproblems, which inherit main new nonstandard features of the full mathematical model. Some results of numerical experiments are discussed.


Models Numerical Topics
Suppose that S 2 = S 22 ∪ S 21 , where are strips that consist of the active and inactive in reaction adsorption sites, respectively.

Chemical Reactions
Let desorbed reaction products of concentrations p 1 (t, x), p 2 (t, x), and p 3 (t, x) diffuse in the same domain.
Let the surface S 1 is impermeable to molecules of reactants and products.
The mechanism of the NO+CO reaction is based on the NO reduction reaction by CO via transformation of the adsorbed product N 2 O and involves the following steps: u j2 and u j1 , j = 1, . . ., 5, are densities of the active and inactive in reaction adsorption sites that are occupied by molecules of adsorbed reactants A 1 and A 2 : NOS (j = 1) and COS (j = 2) and molecules of products NS (j = 3), OS (j = 4), N 2 OS (j = 5).

Models Numerical Topics
Numerical Methods for Surface Reactions Models Numerical Topics

Numerical Integration of ODEs
We should preserve the main properties of the solution: 1. Nonnegativity

Diffusion
We use the non-classical surface diffusion mechanism (a jump diffusion): 1D case: R. ČIEGIS Numerical Methods for Surface Reactions Models Numerical Topics A parabolic diffusion-convection non-stationary problem Equivalent forms (for smooth solutions): R. ČIEGIS Numerical Methods for Surface Reactions 1.The Finite Volume Method is used to get a conservative scheme.
2. Implicit/explicit linearization is done (iterations for the explicit terms can be applied for the CN or the fully implicit Euler approximations).
3. The monotone upwind approximation for the convection transport term is used.
4. If u j (x, t) = 0, then ∂ ∂t u j ≥ 0 is valid for the differential equation.

R. ČIEGIS
Numerical Methods for Surface Reactions Models Numerical Topics

Conjugation conditions
1. Mass conservation condition (the fluxes are continuous) 2. Jump conditions (the solutions are discontinuous)

R. ČIEGIS Numerical Methods for Surface Reactions
We solve the following linear initial-boundary value parabolic problem

Models Numerical Topics
We approximate it by the discrete scheme U n 0 = 0, U n J = 0, U 0 j = u 0 (x j ), j = 0, . . ., J, where the discrete operator A h is defined as

Theorem
The discrete operator A h is symmetric and positive-definite

Theorem
The discrete scheme (2) is stable and the following stability estimates are valid where the L 2 norm is defined as U 2 = (U, U) and U 2 The BC for the 2D problem (the flux of a 1 component):