A New Projection Method for the Zero Froude Number Shallow Water Equations

For non-zero Froude numbers the shallow water equations are a hyperbolic system of partial differential equations. In the zero Froude number limit, they are of mixed hyperbolic-elliptic type, and the velocity field is subject to a divergence constraint. A new semi-implicit projection method for the zero Froude number shallow water equations is presented. This method enforces the divergence constraint on the velocity field, in two steps. First, the numerical fluxes of an auxiliary hyperbolic system are computed with a standard second order method. Then, these fluxes are corrected by solving two Poisson-type equations. These corrections guarantee that the new velocity field satisfies a discrete form of the above-mentioned divergence constraint. The main feature of the new method is a unified discretization of the two Poisson-type equations, which rests on a Petrov-Galerkin finite element formulation with piecewise bilinear ansatz functions for the unknown variable. This discretization naturally leads to piecewise linear ansatz functions for the momentum components. The projection method is derived from a semi-implicit finite volume method for the zero Mach number Euler equations, which uses standard discretizations for the solution of the Poisson-type equations. The new scheme can be formulated as an approximate as well as an exact projection method. In the former case, the divergence constraint is not exactly satisfied. The "approximateness" of the method can be estimated with an asymptotic upper bound of the velocity divergence at the new time level, which is consistent with the method's second-order accuracy. In the exact projection method, the piecewise linear components of the momentum are employed for the computation of the numerical fluxes of the auxiliary system at the new time level. In order to show the stability of the new projection step, a primal-dual mixed finite element formulation is derived, which is equivalent to the Poisson-type equations of the new scheme. Using the abstract theory of Nicola"ides (1982) for generalized saddle point problems, existence and uniqueness of the continuous problem are proven. Furthermore, preliminary results regarding the stability of the discrete method are presented. The numerical results obtained with the new exact method show significant accuracy improvements over the version that uses standard discretizations for the solution of the Poisson-type equations. In the L-two as well as the L-infinity norm, the global error is about four times smaller for smooth solutions. Simulating the advection of a vortex with discontinuous vorticity field, the new method yields a more accurate position of the center of the vortex

Stability of a Cartesian grid projection method for zero Froude number shallow water flows

In this paper a Godunov-type projection method for computing approximate solutions of the zero Froude number (incompressible) shallow water equations is presented. It is second-order accurate and locally conserves height (mass) and momentum. To enforce the underlying divergence constraint on the velocity field, the predicted numerical fluxes, computed with a standard second order method for hyperbolic conservation laws, are corrected in two steps. First, a MAC-type projection adjusts the advective velocity divergence. In a second projection step, additional momentum flux corrections are computed to obtain new time level cell-centered velocities, which satisfy another discrete version of the divergence constraint. The scheme features an exact and stable second projection. It is obtained by a Petrov-Galerkin finite element ansatz with piecewise bilinear trial functions for the unknown incompressible height and piecewise constant test functions. The stability of the projection is proved using the theory of generalized mixed finite elements, which goes back to Nicolaïdes (1982). In order to do so, the validity of three different inf-sup conditions has to be shown. Since the zero Froude number shallow water equations have the same mathematical structure as the incompressible Euler equations of isentropic gas dynamics, the method can be easily transfered to the computation of incompressible variable density flow problems

Stability of a Cartesian grid projection method for zero Froude number shallow water flows

In this paper a Godunov-type projection method for computing approximate solutions of the zero Froude number (incompressible) shallow water equations is presented. It is second-order accurate and locally conserves height (mass) and momentum. To enforce the underlying divergence constraint on the velocity field, the predicted numerical fluxes, computed with a standard second order method for hyperbolic conservation laws and applied to an auxiliary system, are corrected in two steps. First, a MAC-type projection adjusts the advective velocity divergence. In a second projection step, additional momentum flux corrections are computed to obtain new time level cell-centered velocities, which satisfy another discrete version of the divergence constraint. The scheme features an exact and stable second projection. It is obtained by a Petrov-Galerkin finite element ansatz with piecewise bilinear trial functions for the unknown height and piecewise constant test\ud functions. The key innovation compared to existing finite volume projection methods is a correction of the in-cell slopes of the momentum by the second projection. The stability of the projection is proved using a generalized theory for mixed finite elements. In order to do so, the validity of three different inf-sup conditions has to be shown. The results of preliminary numerical test cases demonstrate the method's applicability. On fixed grids the accuracy is improved by a factor four compared to a previous version of the scheme

A Semi-Implicit Multiscale Scheme for Shallow Water Flows at Low Froude Number

A new large time step semi-implicit multiscale method is presented for the solution of low Froude-number shallow water flows. While on small scales which are under-resolved in time the impact of source terms on the divergence of the flow is essentially balanced, on large resolved scales the scheme propagates free gravity waves with minimized diffusion. The scheme features a scale decomposition based on multigrid ideas. Two different time integrators are blended at each scale depending on the scale-dependent Courant number for gravity wave propagation. The finite-volume discretization is based on a Cartesian grid and is second order accurate. The basic properties of the method are validated by numerical tests. This development is a further step in the development of asymptotically adaptive numerical methods for the computation of large scale atmospheric flows

Scale-selective time integration for Long-Wave Linear Acoustics

In this note, we present a new method for the numerical integration of one dimensional linear acoustics with long time steps. It is based on a scale-wise decomposition of the data using standard multigrid ideas and a scale-dependent blending of basic time integrators with different principal features. This enables us to accurately compute balanced solutions with slowly varying short-wave source terms. At the same time, the method effectively filters freely propagating compressible short-wave modes. The selection of the basic time integrators is guided by their discrete-dispersion relation. Furthermore, the ability of the schemes to reproduce balanced solutions is shortly investigated. The method is meant to be used in semi-implicit finite volume methods for weakly compressible flows

A Scale-selective Multilevel Method for Long-Wave Linear Acoustics

A new method for the numerical integration of the equations for one-dimensional linear acoustics with large time steps is presented. While it is capable of computing the "slaved" dynamics of short-wave solution components induced by slow forcing, it eliminates freely propagating compressible short-wave modes, which are under-resolved in time. Scale-wise decomposition of the data based on geometric multigrid ideas enables a scale-dependent blending of time integrators with different principal features. To guide the selection of these integrators, the discrete-dispersion relations of some standard second-order schemes are analyzed, and their response to high wave number low frequency source terms are discussed. The performance of the new method is illustrated on a test case with "multiscale" initial data and a problem with a slowly varying high wave number source term

Mild Type 2 Diabetes Mellitus Reduces the Susceptibility of the Heart to Ischemia/Reperfusion Injury: Identification of Underlying Gene Expression Changes.

Despite clinical studies indicating that diabetic hearts are more sensitive to ischemia/reperfusion injury, experimental data is contradictory. Although mild diabetes prior to ischemia/reperfusion may induce a myocardial adaptation, further research is still needed. Nondiabetic Wistar (W) and type 2 diabetic Goto-Kakizaki (GK) rats (16-week-old) underwent 45 min occlusion of the left anterior descending coronary artery and 24 h reperfusion. The plasma glucose level was significantly higher in diabetic rats compared to the nondiabetics. Diabetes mellitus was associated with ventricular hypertrophy and increased interstitial fibrosis. Inducing myocardial infarction increased the glucose levels in diabetic compared to nondiabetic rats. Furthermore, the infarct size was smaller in GK rats than in the control group. Systolic and diastolic functions were impaired in W + MI and did not reach statistical significance in GK + MI animals compared to the corresponding controls. Among the 125 genes surveyed, 35 genes showed a significant change in expression in GK + MI compared to W + MI rats. Short-term diabetes promotes compensatory mechanisms that may provide cardioprotection against ischemia/reperfusion injury, at least in part, by increased antioxidants and the upregulation of the prosurvival PI3K/Akt pathway, by the downregulation of apoptotic genes, proinflammatory cytokine TNF-alpha, profibrogenic TGF-beta, and hypertrophic marker alpha-actin-1

Linked 3-D modelling of megathrust earthquake-tsunami events: from subduction to tsunami run up

How does megathrust earthquake rupture govern tsunami behaviour? Recent modelling advances permit evaluation of the influence of 3-D earthquake dynamics on tsunami genesis, propagation, and coastal inundation. Here, we present and explore a virtual laboratory in which the tsunami source arises from 3-D coseismic seafloor displacements generated by a dynamic earthquake rupture model. This is achieved by linking open-source earthquake and tsunami computational models that follow discontinuous Galerkin schemes and are facilitated by highly optimized parallel algorithms and software. We present three scenarios demonstrating the flexibility and capabilities of linked modelling. In the first two scenarios, we use a dynamic earthquake source including time-dependent spontaneous failure along a 3-D planar fault surrounded by homogeneous rock and depth-dependent, near-lithostatic stresses. We investigate how slip to the trench influences tsunami behaviour by simulating one blind and one surface-breaching rupture. The blind rupture scenario exhibits distinct earthquake characteristics (lower slip, shorter rupture duration, lower stress drop, lower rupture speed), but the tsunami is similar to that from the surface-breaching rupture in run-up and length of impacted coastline. The higher tsunami-generating efficiency of the blind rupture may explain how there are differences in earthquake characteristics between the scenarios, but similarities in tsunami inundation patterns. However, the lower seafloor displacements in the blind rupture result in a smaller displaced volume of water leading to a narrower inundation corridor inland from the coast and a 15 per cent smaller inundation area overall. In the third scenario, the 3-D earthquake model is initialized using a seismo-thermo-mechanical geodynamic model simulating both subduction dynamics and seismic cycles. This ensures that the curved fault geometry, heterogeneous stresses and strength and material structure are consistent with each other and with millions of years of modelled deformation in the subduction channel. These conditions lead to a realistic rupture in terms of velocity and stress drop that is blind, but efficiently generates a tsunami. In all scenarios, comparison with the tsunamis sourced by the time-dependent seafloor displacements, using only the time-independent displacements alters tsunami temporal behaviour, resulting in later tsunami arrival at the coast, but faster coastal inundation. In the scenarios with the surface-breaching and subduction-initialized earthquakes, using the time-independent displacements also overpredicts run-up. In the future, the here presented scenarios may be useful for comparison of alternative dynamic earthquake-tsunami modelling approaches or linking choices, and can be readily developed into more complex applications to study how earthquake source dynamics influence tsunami genesis, propagation and inundation