arXiv subjects

Christopher Zoppou

Publications and source records attributed to Christopher Zoppou.

5 recordsLinked to original sources

arXiv2023

Strongly stable dual-pairing summation by parts finite difference schemes for the vector invariant nonlinear shallow water equations -- I: Numerical scheme and validation on the plane

We present an energy/entropy stable and high order accurate finite difference (FD) method for solving the nonlinear (rotating) shallow water equations (SWEs) in vector invariant form using the newly developed dual-pairing and dispersion-relation preserving summation by parts (SBP) FD operators. We derive new well-posed boundary conditions (BCs) for the SWE in one space dimension, formulated in terms of fluxes and applicable to linear and nonlinear SWEs. For the nonlinear vector invariant SWE in the subcritical regime, where energy is an entropy functional, we find that energy/entropy stability ensures the boundedness of numerical solution but does not guarantee convergence. Adequate amount of numerical dissipation is necessary to control high frequency errors which could negatively impact accuracy in the numerical simulations. Using the dual-pairing SBP framework, we derive high order accurate and nonlinear hyper-viscosity operator which dissipates entropy and enstrophy. The hyper-viscosity operator effectively minimises oscillations from shocks and discontinuities, and eliminates high frequency grid-scale errors. The numerical method is most suitable for the simulations of subcritical flows typically observed in atmospheric and geostrophic flow problems. We prove both nonlinear and local linear stability results, as well as a priori error estimates for the semi-discrete approximations of both linear and nonlinear SWEs. Convergence, accuracy, and well-balanced properties are verified via the method of manufactured solutions and canonical test problems such as the dam break and lake at rest. Numerical simulations in two-dimensions are presented which include the rotating and merging vortex problem and barotropic shear instability, with fully developed turbulence.

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arXiv2023

On well-posed boundary conditions and energy stable finite volume method for the linear shallow water wave equation

We derive and analyse well-posed boundary conditions for the linear shallow water wave equation. The analysis is based on the energy method and it identifies the number, location and form of the boundary conditions so that the initial boundary value problem is well-posed. A finite volume method is developed based on the summation-by-parts framework with the boundary conditions implemented weakly using penalties. Stability is proven by deriving a discrete energy estimate analogous to the continuous estimate. The continuous and discrete analysis covers all flow regimes. Numerical experiments are presented verifying the analysis.

math.NA↗

arXiv2023

Well-posed boundary conditions and energy stable discontinuous Galerkin spectral element method for the linearized Serre equations

We derive well-posed boundary conditions for the linearized Serre equations in one spatial dimension by utilizing the energy method. An energy stable and conservative discontinuous Galerkin spectral element method with simple upwind numerical fluxes is proposed for solving the initial boundary value problem. We derive discrete energy estimates for the numerical approximation and prove a priori error estimates in the energy norm. Detailed numerical examples are provided to verify the theoretical analysis and show convergence of numerical errors.

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arXiv2017

Behaviour of the Serre Equations in the Presence of Steep Gradients Revisited

We use numerical methods to study the behaviour of the Serre equations in the presence of steep gradients because there are no known analytical solutions for these problems. In keeping with the literature we study a class of initial condition problems that are a smooth approximation to the initial conditions of the dam-break problem. This class of initial condition problems allow us to observe the behaviour of the Serre equations with varying steepness of the initial conditions. The numerical solutions of the Serre equations are justified by demonstrating that as the resolution increases they converge to a solution with little error in conservation of mass, momentum and energy independent of the numerical method. We observe four different structures of the converged numerical solutions depending on the steepness of the initial conditions. Two of these structures were observed in the literature, with the other two not being commonly found in the literature. The numerical solutions are then used to assess how well the analytical solution of the shallow water wave equations captures the mean behaviour of the solution of the Serre equations for the dam-break problem. Lastly the numerical solutions are used to evaluate the usefulness of asymptotic results in the literature to approximate the depth and location of the front of an undular bore.

math.NA↗

arXiv2016

Third-order Finite Volume/Finite Element Solution of the Fully Nonlinear Weakly Dispersive Serre Equations

The nonlinear weakly dispersive Serre equations contain higher-order dispersive terms. This includes a mixed derivative flux term which is difficult to handle numerically. The mix spatial and temporal derivative dispersive term is replaced by a combination of temporal and spatial terms. The Serre equations are re-written so that the system of equations contain homogeneous derivative terms only. The reformulated Serre equations involve the water depth and a new quantity as the conserved variables which are evolved using the finite volume method. The remaining primitive variable, the velocity is obtained by solving a second-order elliptic equation using the finite element method. To avoid the introduction of numerical dispersion that may dominate the physical dispersion, the hybrid scheme has third-order accuracy. Using analytical solutions, laboratory flume data and by simulating the dam-break problem, the proposed scheme is shown to be accurate, simple to implement and stable for a range of problems, including discontinuous flows.

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