Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations

Abstract

We first analyse a semi-discrete operator splitting method for nonlinear, possibly strongly degenerate, convection-diffusion equations. Due to strong degeneracy, solutions can be discontinuous and are in general not uniquely determined by their data. Hence weak solutions satisfying an entropy condition are sought. We then propose and analyse a fully discrete splitting method which employs a front tracking scheme for the convection step and a finite difference scheme for the diffusion step. Numerical examples are presented which demonstrate that our method can be used to compute physically correct solutions to mixed hyperbolic-parabolic convection-diffusion equations.Mathematics Subject Classification (1991): 65M12, 35K65, 35L65

Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients

“…The (very recent) literature include papers by Evje and Karlsen [18], Holden et al [25], and Holden et al [26] on operator splitting methods (see also the lecture notes by Espedal and Karlsen [15]); Evje and Karlsen [16,17,19,20] on upwind difference schemes; Kurganov and Tadmor [35] on central difference schemes; Bouchut et al [2] on kinetic BGK schemes; Afif and Amaziane [1] and Ohlberger [39] on finite volume methods; and Cockburn and Shu [11] on the local discontinuous Galerkin method. Strictly speaking, the authors of [1,11,35] do not analyze their numerical methods within an entropy solution framework.…”

Operator Splitting Methods for Generalized Korteweg–De Vries Equations

“…An unconditionally stable splitting scheme for the equation with F = εu xx + g(t, x, u) was analyzed in [12]. Finally, Evje and Karlsen [5] treated the case with a possibly degenerate viscous term F = (a(u)u x ) x , where a may vanish, using operator splitting. There is an important difference 2 between the diffusive or viscous case…”

Corrected Operator Splitting for Nonlinear Parabolic Equations

“…The mixed hyperbolic/parabolic case (d(u) ≥ 0) is addressed in [14]. In the parabolic context we can obviously write (2) in conservative form (1), so that any solution strategy presented for (1) applies equally as well to (2).…”

“…A widely used strategy is viscous OS, that is, splitting (1) into a hyperbolic conservation law and a parabolic heat equation, each of which is solved by some proper numerical scheme. This approach, or at least certain variations on this approach, has indeed been taken by several authors; we mention Beale and Majda [2], Douglas and Russell [10], Russell [31], Ewing and Russell [13], Espedal and Ewing [11], Ewing [12], Dahle [9], Dawson [8], Karlsen and Risebro [20], and more recently Evje and Karlsen [14]. In [31], a characteristic element method is used to solve the hyperbolic part of (1).…”

“…In [20] it is shown that the viscous splitting method converges to the solution of (1) (also in multidimensions) in the case of linear diffusion, a Lipschitz continuous flux function, and any (discontinuous) initial function of bounded variation. Convergence results for the viscous splitting method for one-dimensional parabolic equations with a nonlinear, possibly strongly degenerate, diffusion term are obtained in [14].…”