Riemann solvers form a crucial component in the numerical solution of partial differential equations (PDEs) using Discontinuous Galerkin (DG) or Finite Volumes methods. This thesis implements HLL-type, exact, and enhanced Riemann solvers in the open-source software ExaHyPE~2, an engine to generate code for simulating hyperbolic PDEs in first-order formulation. The solvers are evaluated and verified with the ADER-DG approach applied to the non-homogeneous shallow water and linear elastic wave equations in various example problems like dam break, lake at rest, and oscillating lake. The work shows that in this context, the choice of Riemann solver influences properties such as accuracy or well-balancedness. However, due to the high order of convergence, the differences between the solvers are small, particularly for scenarios with smooth solutions. Furthermore, the necessary adjustments to the solvers to allow for numerically demanding problems like inundation and tsunami simulations are outlined.