3rd Runge-Kutta
 

Hello everyone

What is the benefits of using the 3rd explicit Runge-Kutta Schem for the convective term of the Navier-Stokes equations in terms of the time advancement

if you read about the Modified Equation construction, you can see that errors from time integration can actually map into spatial error effects. Using higher-order time advancement schemes (whether RK or Adams-type schemes) will tend to decrease spatial errors, especially numerical diffusion. RK schemes in particular are also very effective because they tend to be more stable (hence, allow larger time steps) then similar ordered explicit Adams methods. Also, a lot of work has been done to mate RK schemes with different nonlinear spatial discretizations/flux limiters and analyze the resulting Total-Variation Boundary/Diminishing properties. TVD/TVB properties can be destroyed by poor choice of time integration schemes.

Treating convective terms with implicit schemes would be hard due to non-linearity. So convective terms are ( to my knowledge ) often treated with explicit schemes. And it seems like RK (Runge-Kutta) schemes, which are multi-stage, outshine multi-step AB(Adams–Bashforth) schemes due to stability properties which allows you take larger time steps. ( actually computational cost wise AB2-RK3 are similar when you look at the clocks per physical simulation time but then you gain 1 extra order of accuracy due to smartly selected sub_delta t steps.)

Also when it comes to starting you simulation from a given velocity field, its a benefit to have a RK scheme since AB would need 2 time level initialization.

For atmospheric boundary layer simulations in my area treating convective terms with low storage RK3/RK4 is very popular.

To my knowledge, RK methods are popular when the semi-discrete methods are used, that is the PDE system is first discretized in space reducing to a system of ODEs.