Research topics:

Main problems:

1. Generalization of analysis for additive RK methods with different step-sizes (\(\tau, \tilde{\tau}\)) to multistep methods.

2. Solving the implicit downwind problem. How to maintain the positivity of the solution using an iterative solver on a downwind scheme (Waveform relaxation)?

3. Optimality of downwind methods. For a given RK method what is the best choice of \(\tilde{A}\) that maximize the CFL coefficient?

Smaller problems:

1. Find algebraic expressions for the coefficients of implicit high order downwind schemes.

2. High order explicit downwind methods (\(p>5\), Ruuth-Spiteri method).

3. Linear analysis for DWRK methods. Convergence analysis - upper bounds for global error. Extension of the analysis to additive methods.

4. Analysis of boundary conditions and their influence on downwind methods. How to impose correctly the boundary conditions?

5. Study of the radius of absolute monotonicity of two variables polynomials.