##
**Godunov-mixed methods for advection-diffusion equations in multidimensions.**
*(English)*
Zbl 0791.65062

The author considers time-split methods for multidimensional advection-diffusion equations. The advection is approximated by a Godunov-type procedure while diffusion is approximated by a low-order mixed finite element method.

Section 2 states the problem of interest and develops the basic algorithm. The set-up is confined to \(\mathbb{R}^ 2\), however the analysis can be easily extended to \(\mathbb{R}^ 3\). The main problem reads as follows: let \(s({\mathbf x},t)\) satisfy (1) \(s_ t+f({\mathbf x},t,s)_ x+g({\mathbf x},t,s)_ y-\nabla (D ({\mathbf x},t) \nabla s)=0\) on \(\Omega \times (0,T]\), (2) \(s=b\) on \(\partial \Omega \times(0,T]\), (3) \(s=s^ 0\) on \(\Omega \times \{0\}\), where \({\mathbf x}=(x,y)\) and \(\Omega\) is a convex bounded polygon in \(\mathbb{R}^ 2\). Then the basic method is described and an error estimate is proved (Theorem 2.1). The proof relies on Gronwall’s Lemma and a standard inequality of type \(ab \leq {\varepsilon \over 2} a^ 2+{2 \over \varepsilon} b^ 2\). It is worthy to underline that Theorem 2.1 does not explicitly require any assumptions on the relative sizes of \(\Delta t\) and \(h\) (time and spatial step, respectively), what in many applications may be advantageous.

In the next three sections three special cases are discussed and dealt with using this basic theorem. In the first approach (§3) advective fluxes are approximated by an unsplit higher order Godunov procedure. A rectangular triangulation of \(\overline \Omega\) is assumed. The method turns out to be first-order accurate in time and second-order accurate in space. In §4 there is a modification of this approach which is of second-order in time. The modification consists of using Crank-Nicolson time-stepping, and adding a term to the left and right states at each element edge.

The last method, described in §5, is based on calculation of fluxes by characteristic tracing over, potentially more than one element. Here triangular elements are assumed. Using this method first-order accuracy in time and space is gained for linear problems, i.e. \(f({\mathbf x},s) = u({\mathbf x})s\) and \(g({\mathbf x},s) = v({\mathbf x})s\) in the problem (1), (2), (3).

Section 2 states the problem of interest and develops the basic algorithm. The set-up is confined to \(\mathbb{R}^ 2\), however the analysis can be easily extended to \(\mathbb{R}^ 3\). The main problem reads as follows: let \(s({\mathbf x},t)\) satisfy (1) \(s_ t+f({\mathbf x},t,s)_ x+g({\mathbf x},t,s)_ y-\nabla (D ({\mathbf x},t) \nabla s)=0\) on \(\Omega \times (0,T]\), (2) \(s=b\) on \(\partial \Omega \times(0,T]\), (3) \(s=s^ 0\) on \(\Omega \times \{0\}\), where \({\mathbf x}=(x,y)\) and \(\Omega\) is a convex bounded polygon in \(\mathbb{R}^ 2\). Then the basic method is described and an error estimate is proved (Theorem 2.1). The proof relies on Gronwall’s Lemma and a standard inequality of type \(ab \leq {\varepsilon \over 2} a^ 2+{2 \over \varepsilon} b^ 2\). It is worthy to underline that Theorem 2.1 does not explicitly require any assumptions on the relative sizes of \(\Delta t\) and \(h\) (time and spatial step, respectively), what in many applications may be advantageous.

In the next three sections three special cases are discussed and dealt with using this basic theorem. In the first approach (§3) advective fluxes are approximated by an unsplit higher order Godunov procedure. A rectangular triangulation of \(\overline \Omega\) is assumed. The method turns out to be first-order accurate in time and second-order accurate in space. In §4 there is a modification of this approach which is of second-order in time. The modification consists of using Crank-Nicolson time-stepping, and adding a term to the left and right states at each element edge.

The last method, described in §5, is based on calculation of fluxes by characteristic tracing over, potentially more than one element. Here triangular elements are assumed. Using this method first-order accuracy in time and space is gained for linear problems, i.e. \(f({\mathbf x},s) = u({\mathbf x})s\) and \(g({\mathbf x},s) = v({\mathbf x})s\) in the problem (1), (2), (3).

Reviewer: S.Burys (Kraków)

### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

35K55 | Nonlinear parabolic equations |