Solving the Shallow Water Equations Using the High Order Space-Time Discontinuous Galerkin Cell-Vertex Scheme

2012 ◽  
Author(s):  
Shuangzhang Tu ◽  
Qing Pang ◽  
Haibin Xiang
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Shallow Water Equations ◽  
Space Time ◽  
High Order

High-order discontinuous Galerkin schemes on general 2D manifolds applied to the shallow water equations

2009 ◽  
Vol 228 (17) ◽  
pp. 6514-6535 ◽  
Author(s):  
P.-E. Bernard ◽  
J.-F. Remacle ◽  
R. Comblen ◽  
V. Legat ◽  
K. Hillewaert
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Shallow Water Equations ◽  
High Order

An adaptive space-time shock capturing method with high order wavelet bases for the system of shallow water equations

2018 ◽  
Vol 28 (12) ◽  
pp. 2842-2861
Author(s):  
Hadi Minbashian ◽  
Hojatollah Adibi ◽  
Mehdi Dehghan
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Degrees Of Freedom ◽  
Adaptive Method ◽  
Space Time ◽  
High Order ◽  
Shock Capturing ◽  
Weak Formulation ◽  
Content Type ◽  
Wavelet Expansions

PurposeThis paper aims to propose an adaptive method for the numerical solution of the shallow water equations (SWEs). The authors provide an arbitrary high-order method using high-order spline wavelets. Furthermore, they use a non-linear shock capturing (SC) diffusion which removes the necessity of post-processing.Design/methodology/approachThe authors use a space-time weak formulation of SWEs which exploits continuous Galerkin (cG) in space and discontinuous Galerkin (dG) in time allowing time stepping, also known as cGdG. Such formulations along with SC term have recently been proved to ensure the stability of fully discrete schemes without scarifying the accuracy. However, the resulting scheme is expensive in terms of number of degrees of freedom (DoFs). By using natural adaptivity of wavelet expansions, the authors devise an adaptive algorithm to reduce the number of DoFs.FindingsThe proposed algorithm uses DoFs in a dynamic way to capture the shocks in all time steps while keeping the representation of approximate solution sparse. The performance of the proposed scheme is shown through some numerical examples.Originality/valueAn incorporation of wavelets for adaptivity in space-time weak formulations applied for SWEs is proposed.

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