Abstract
In this article, fifth order well-balanced finite volume multi-resolution weighted essentially non-oscillatory (FV MR-WENO) schemes are constructed for solving one-dimensional and two-dimensional Ripa models. The Ripa system generalizes the shallow water model by incorporating horizontal temperature gradients. The presence of temperature gradients and source terms in the Ripa models introduce difficulties in developing high order accurate numerical schemes which can preserve exactly the steady-state conditions. The proposed numerical methods are capable to exactly preserve the steady-state solutions and maintain non-oscillatory property near the shock transitions. Moreover, in the procedure of derivation of the FV MR-WENO schemes unequal central spatial stencils are used and linear weights can be chosen any positive numbers with only restriction that their total sum is one. Various numerical test problems are considered to check the validity and accuracy of the derived numerical schemes. Further, the results obtained from considered numerical schemes are compared with those of a high resolution central upwind scheme and available exact solutions of the Ripa model.
Highlights
The shallow water equations (SWEs) are of great importance due to wide range applications in incompressible flows, such as bore propagation, solute transport, currents in estuaries, and in surges or tsunamis phenomenon
This shows that the study of Ripa models is of great importance to understand the various realworld phenomenon
The main objectives of the suggested numerical schemes are to preserve the steady-state solutions without sacrificing the high order accuracy and do not create unwanted oscillations in the vicinity of temperature jump
Summary
The shallow water equations (SWEs) are of great importance due to wide range applications in incompressible flows, such as bore propagation, solute transport, currents in estuaries, and in surges or tsunamis phenomenon. Fifth order well-balanced finite volume multi-resolution WENO schemes are constructed for solving the one-dimensional (1D) and two-dimensional (2D) Ripa models with and without source terms. The main objectives of the suggested numerical schemes are to preserve the steady-state solutions without sacrificing the high order accuracy and do not create unwanted oscillations in the vicinity of temperature jump This objective is achieved by decomposing the integral of source terms into the sum of particular terms, computing each term in a way which is consistent to the computation of corresponding numerical fluxes. Similar to the 1D case, for the steady-state in equation (11), the source terms are completely balanced by the flux gradients in 2D Ripa model.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
