Development of algorithms for convection-dominated problemsundercompressible flows is an important topic for applications like sediment transport.If the transport of a quantity saturates then the solution of the transport equations must be bounded by physical constraints. This process could in principle be modeled via the constitutive relations of the flow model. Alternatively, one could impose algebraicconstraints in the transport solver to guarantee the physical bounds.With this project we aim to review recent methods for solving transport equations (under compressible flows) that impose physically motivated bounds on the solution.In addition, we are interested on exploring novel methodologies based on flux correctionfor continuous Galerkin finite elements to achieve the desired goals.​​