High Order Well-Balanced Weighted Compact Nonlinear Schemes for Shallow Water Equations

AbstractIn this study, a numerical framework of the high order well-balanced weighted compact nonlinear (WCN) schemes is proposed for the shallow water equations based on the work in [S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008) 7294-7321]. We employ a special splitting technique for the source term proposed in [Y. Xing, C.-W Shu, J. Comput. Phys. 208 (2005) 206-227] to maintain the exact C-property, which can be proved theoretically. In the meantime, the genuine high order accuracy of the numerical scheme can be observed successfully, and small perturbation of the stationary state can be resolved and evolved well. In order to capture the strong discontinuities and large gradients, the fifth-order upwind weighted nonlinear interpolations together with the fourth/sixth order cell-centered compact scheme are used to construct different WCN schemes. In addition, the local characteristic projections are considered to further restrain the potential numerical oscillations. A variety of representative one- and two-dimensional examples are tested to demonstrate the good performance of the proposed schemes.

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High Order Well-Balanced Weighted Compact Nonlinear Schemes for the Gas Dynamic Equations under Gravitational Fields

East Asian Journal on Applied Mathematics ◽

10.4208/eajam.181016.300517g ◽

2017 ◽

Vol 7 (4) ◽

pp. 697-713

Author(s):

Zhen Gao ◽

Guanghui Hu

Keyword(s):

Steady State Solution ◽

High Order ◽

Dynamic Equations ◽

High Order Accuracy ◽

Local Characteristic ◽

Gravitational Fields ◽

Strong Discontinuities ◽

Gas Dynamic ◽

Coarse Meshes ◽

Gas Dynamic Equations

AbstractIn this study, we propose a high order well-balanced weighted compact nonlinear (WCN) scheme for the gas dynamic equations under gravitational fields. The proposed scheme is an extension of the high order WCN schemes developed in (S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008) 7294-7321). For the purpose of maintaining the exact steady state solution, the well-balanced technique in (Y. Xing, C.-W Shu, J. Sci. Comput. 54 (2013) 645-662) is employed to split the source term into two terms. The proposed scheme can maintain the isothermal equilibrium solution exactly, genuine high order accuracy and resolve small perturbations of the hydrostatic balance state on the coarse meshes. Furthermore, in order to capture the strong discontinuities and large gradients, the fifth-order upwind weighted nonlinear interpolations together with the fourth/sixth order cell-centered compact schemes with local characteristic projections are used to construct different WCN schemes. Several representative one- and two-dimensional examples are simulated to demonstrate the good performance of the proposed schemes.

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A Fifth-Order Combined Compact Difference Scheme for Stokes Flow on Polar Geometries

East Asian Journal on Applied Mathematics ◽

10.4208/eajam.200816.300517a ◽

2017 ◽

Vol 7 (4) ◽

pp. 714-727 ◽

Cited By ~ 5

Author(s):

Dongdong He ◽

Kejia Pan

Keyword(s):

Differential Equations ◽

Stream Function ◽

Stokes Flow ◽

Incompressible Flows ◽

Stokes Equations ◽

High Order ◽

High Order Accuracy ◽

Order Accuracy ◽

Compact Difference ◽

Fifth Order

AbstractIncompressible flows with zero Reynolds number can be modeled by the Stokes equations. When numerically solving the Stokes flow in stream-vorticity formulation with high-order accuracy, it will be important to solve both the stream function and velocity components with the high-order accuracy simultaneously. In this work, we will develop a fifth-order spectral/combined compact difference (CCD) method for the Stokes equation in stream-vorticity formulation on the polar geometries, including a unit disk and an annular domain. We first use the truncated Fourier series to derive a coupled system of singular ordinary differential equations for the Fourier coefficients, then use a shifted grid to handle the coordinate singularity without pole condition. More importantly, a three-point CCD scheme is developed to solve the obtained system of differential equations. Numerical results are presented to show that the proposed spectral/CCD method can obtain all physical quantities in the Stokes flow, including the stream function and vorticity function as well as all velocity components, with fifth-order accuracy, which is much more accurate and efficient than low-order methods in the literature.

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A high-order conservative collocation scheme and its application to global shallow water equations

Geoscientific Model Development Discussions ◽

10.5194/gmdd-7-4251-2014 ◽

2014 ◽

Vol 7 (4) ◽

pp. 4251-4290 ◽

Cited By ~ 1

Author(s):

C. Chen ◽

X. Li ◽

X. Shen ◽

F. Xiao

Keyword(s):

Shallow Water ◽

General Circulation ◽

Control Volume ◽

High Order ◽

High Order Accuracy ◽

Water Model ◽

Shallow Water Model ◽

Order Accuracy ◽

Flux Function ◽

Dependent Variables

Abstract. An efficient and conservative collocation method is proposed and used to develop a global shallow water model in this paper. Being a nodal type high-order scheme, the present method solves the point-wise values of dependent variables as the unknowns within each control volume. The solution points are arranged as Gauss–Legendre points to achieve the high-order accuracy. The time evolution equations to update the unknowns are derived under the flux-reconstruction (FR) framework (Huynh, 2007). Constraint conditions used to build the spatial reconstruction for the flux function include the point-wise values of flux function at the solution points, which are computed directly from the dependent variables, as well as the numerical fluxes at the boundaries of the control volume which are obtained as the Riemann solutions between the adjacent cells. Given the reconstructed flux function, the time tendencies of the unknowns can be obtained directly from the governing equations of differential form. The resulting schemes have super convergence and rigorous numerical conservativeness. A three-point scheme of fifth-order accuracy is presented and analyzed in this paper. The proposed scheme is adopted to develop the global shallow-water model on the cubed-sphere grid where the local high-order reconstruction is very beneficial for the data communications between adjacent patches. We have used the standard benchmark tests to verify the numerical model, which reveals its great potential as a candidate formulation for developing high-performance general circulation models.

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A high-order conservative collocation scheme and its application to global shallow-water equations

Geoscientific Model Development ◽

10.5194/gmd-8-221-2015 ◽

2015 ◽

Vol 8 (2) ◽

pp. 221-233 ◽

Cited By ~ 1

Author(s):

C. Chen ◽

X. Li ◽

X. Shen ◽

F. Xiao

Keyword(s):

Shallow Water ◽

General Circulation ◽

General Circulation Models ◽

High Order ◽

High Order Accuracy ◽

Water Model ◽

Shallow Water Model ◽

Order Accuracy ◽

Flux Function ◽

Dependent Variables

Abstract. In this paper, an efficient and conservative collocation method is proposed and used to develop a global shallow-water model. Being a nodal type high-order scheme, the present method solves the pointwise values of dependent variables as the unknowns within each control volume. The solution points are arranged as Gauss–Legendre points to achieve high-order accuracy. The time evolution equations to update the unknowns are derived under the flux reconstruction (FR) framework (Huynh, 2007). Constraint conditions used to build the spatial reconstruction for the flux function include the pointwise values of flux function at the solution points, which are computed directly from the dependent variables, as well as the numerical fluxes at the boundaries of the computational element, which are obtained as Riemann solutions between the adjacent elements. Given the reconstructed flux function, the time tendencies of the unknowns can be obtained directly from the governing equations of differential form. The resulting schemes have super convergence and rigorous numerical conservativeness. A three-point scheme of fifth-order accuracy is presented and analyzed in this paper. The proposed scheme is adopted to develop the global shallow-water model on the cubed-sphere grid, where the local high-order reconstruction is very beneficial for the data communications between adjacent patches. We have used the standard benchmark tests to verify the numerical model, which reveals its great potential as a candidate formulation for developing high-performance general circulation models.

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