Error Driven Node Placement as Applied to One Dimensional Shallow Water Equations

This paper investigates the technique of localised mesh refinement to solve the Shallow Water Equations (SWE), by adding nodes only where needed. The discretization process linearizes the nonlinear equations for solving as a linear system. The nonlinear error values at specific nodes are used to indicate which node will have additional nodes added either side. The process of adding nodes is repeated until the nonlinear error value is below a given threshold, or the predefined maximum number of nodes for that given time step has been reached. This process is restarted again at each time step, allowing the optimization process to efficiently allocate nodes based only on error, avoiding global increases in node numbers across the solution set.