Janusz Pudykiewicz
In order to provide an alternative to semi-Lagrangian techniques for the solution of the
reaction-advection-diffusion equation, a simple yet accurate Eulerian algorithm build upon the principle
of finite volumes has been developed.
The proposed approach exploits the concept of semidiscretization (see Leveque, Finite Volume methods for
hyperbolic problems; page 191); first, the convective and diffusive fluxes in the equation are
approximated and then, the resulting set of the Ordinary Differential Equations (ODEs) is solved using the
appropriate time stepping algorithm. This methodology has been selected because of both its flexibility
with respect to the mesh
selection and its inherent ability to represent subgrid-scale processes and discontinuities in the
solution. The method is initially applied on a geodesic icosahedral grid composed of
triangles with vertices located on the spherical surface.
Depending on the stiffness of the system, the time integration of the set of the equations obtained after
space discretization is performed either with the explicit 4-th order Runge-Kutta scheme or or with the
semi-implicit Runge-Kutta-Rosenbrock solver.
The performance of the advection-diffusion scheme is assessed using the suite of standard tests. The main
conclusion from these tests is that the presented method offers mass conservation, quasi monotonicity and
good accuracy. Also, the algorithm is stable for
the advection with Courant numbers of the order of 2.5.
In order to fully evaluate the method, the reaction-diffusion system on the sphere was also considered.
The comparison of the numerical results to the analytical solution of the reaction-diffusion waves
presented by Turing shows that the method is quite accurate with
the maximum error not exceeding 0.01 percent. The solver was also applied for the integration of the
nonlinear chemical kinetics system known as a Brusselator in order to illustrate the effect of pattern
formation on the sphere.
The main intended application of the reaction-advection-diffusion algorithm presented during the seminar
is the simulation of the chemical constituents in the Earth atmosphere. The presented scheme can be
easily applied to the full system of conservation laws for fluids in arbitrary geometry with both
structured and unstructured meshes.
Before exploring the full potential of the method in the geophysical context the formulation of the solver
for shallow water equations will be presented.