Well-balanced central schemes for the One-dimensional shallow water equations

Abstract

This thesis presents and contrasts two unique Unstaggered Central Scheme (UCS)’s for hyperbolic systems, specifically Shallow Water Equations (SWE): an Unstaggered Central Scheme with the Subraction Method (UCS-Sub) and an Unstaggered Central Weighted Essentially Non-Oscillatory Scheme with the Subtraction Method (UCWENO-Sub). Both schemes are made to protect the hyperbolic systems’ well-balanced (WB) characteristic, which keeps Steady state (SS) solutions immobile. This is made possible by implementing the subtraction method (SM), which effectively removes spurious numerical oscillations. Unstaggered schemes do not require the computationally costly step of solving Riemann problems at cell interfaces, which is a requirement of standard staggered systems. This results in notable efficiency gains. We perform an extensive comparison between UCS-Sub and UCWENO-Sub on several benchmark tasks to assess their respective performances. The outcomes show that both techniques are useful for approximating solutions to hyperbolic systems; while UCWENO-Sub gives priority to higher-order accuracy, they both strike a balance between simplicity, efficiency, and accuracy.