Vincenzo Casulli

An unstructured grid, three-dimensional model based on the shallow water equations

A semi-implicit finite difference model based on the three-dimensional shallow water equations is modified to use unstructured grids. There are obvious advantages in using unstructured grids in problems with a complicated geometry. In this development, the concept of unstructured orthogonal grids is introduced and applied to this model. The governing differential equations are discretized by means of a semi-implicit algorithm that is robust, stable and very efficient. The resulting model is relatively simple, conserves mass, can fit complicated boundaries and yet is sufficiently flexible to permit local mesh refinements in areas of interest. Moreover, the simulation of the flooding and drying is included in a natural and straightforward manner. These features are illustrated by a test case for studies of convergence rates and by examples of flooding on a river plain and flow in a shallow estuary. Copyright

A semi-implicit finite difference method for non-hydrostatic, free-surface flows

In this paper a semi-implicit finite difference model for non-hydrostatic, free-surface flows is analyzed and discussed. It is shown that the present algorithm is generally more accurate than recently developed models for quasi-hydrostatic flows. The governing equations are the free-surface Navier–Stokes equations defined on a general, irregular domain of arbitrary scale. The momentum equations, the incompressibility condition and the equation for the free-surface are integrated by a semi-implicit algorithm in such a fashion that the resulting numerical solution is mass conservative and unconditionally stable with respect to the gravity wave speed, wind stress, vertical viscosity and bottom friction. Copyright

Iterative Solution of Piecewise Linear Systems

The correct formulation of numerical models for free-surface hydrodynamics often requires the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself. In so doing one may prevent the development of unrealistic negative water depths. The resulting piecewise linear systems are equivalent to particular linear complementarity problems whose solutions could be obtained by using, for example, interior point methods. These methods may have a favorable convergence property, but they are purely iterative and convergence to the exact solution is proven only in the limit of an infinite number of iterations. In the present paper a simple Newton-type procedure for certain piecewise linear systems is derived and discussed. This procedure is shown to have a finite termination property, i.e., it converges to the exact solution in a finite number of steps, and, actually, it converges very quickly, as confirmed by a few numerical tests.

Iterative Solution of Piecewise Linear Systems and Applications to Flows in Porous Media

The correct numerical modeling of free-surface hydrodynamic problems often requires to have the solution of special linear systems whose coefficient matrix is a piecewise constant function of the solution itself. In doing so, one may fulfill relevant physical constraints. The existence, the uniqueness, and two constructive iterative methods to solve a piecewise linear system of the form

A staggered semi-implicit spectral discontinuous Galerkin scheme for the shallow water equations

\max[\boldsymbol{l},\min(\boldsymbol{u},\mathbf{x})]+T\mathbf{x}=\mathbf{b}

A semi‐implicit method for vertical transport in multidimensional models

are analyzed. The methods are shown to have a finite termination property; i.e., they converge to an exact solution in a finite number of steps and, actually, they converge very quickly, as confirmed by a few numerical tests, which are derived from the mathematical modeling of flows in porous media.

Semi-implicit numerical modeling of axially symmetric flows in compliant arterial systems

A spatially arbitrary high order, semi-implicit spectral discontinuous Galerkin (DG) scheme for the numerical solution of the shallow water equations on staggered control volumes is derived and discussed. The free surface elevation and the momentum are expressed in terms of the same polynomials that are used as basis, and as test functions. Each unknown is, however, defined on a different set of control volumes that are spatially staggered with respect to each other. A semi-implicit time integration yields a stable and efficient mass conservative algorithm. The use of a staggered mesh has the advantage that after substitution of the momentum equations into the mass conservation equation only one single block-penta-diagonal system must be solved for the new free surface location, which is a scalar quantity. Subsequently the new momentum components are directly obtained. The staggered semi-implicit approach makes the present DG scheme different from other published DG schemes. The sparse linear system of the resulting discrete wave equation for the free surface can be conveniently solved by a matrix-free GMRES algorithm. Furthermore, for the shallow water equations the proposed scheme can be written as a quadrature-free method, where all surface and volume integrals can be precomputed once and for all on the reference element and assembled into universal matrices and tensors. For the special case N=0 the proposed method reduces to a classical semi-implicit finite difference scheme. The proposed semi-implicit scheme is particularly well suited for low Froude number flows. The method is validated on some typical academic benchmark problems, using polynomial degrees of up to N=20.

Evaluation of the UnTRIM model for 3-D tidal circulation

A one-dimensional scalar transport method which is appropriate for simulations over a wide range of Courant number is described. Von Neumann stability and matrix invertibility are guaranteed for all Courant numbers and the method has less diffusive and dispersive error than simpler implicit methods. It is implemented for vertical scalar transport in a three-dimensional hydrodynamic model, with horizontal transport discretized explicitly. The method is applied and compared with simpler semi-implicit methods in several test cases and used for a simulation of scalar transport in an estuary.

Numerical simulation of three-dimensional free surface flow in isopycnal co-ordinates

Blood flow in arterial systems is described by the three-dimensional Navier-Stokes equations within a time-dependent spatial domain that accounts for the viscoelasticity of the arterial walls. These equations are simplified by assuming cylindrical geometry, axially symmetric flow, and hydrostatic equilibrium in the radial direction. In this paper, an efficient semi-implicit method is formulated in such a fashion that numerical stability is obtained at a minimal computational cost. The resulting computer model is relatively simple, robust, accurate, and extremely efficient. These features are illustrated on nontrivial test cases where the exact analytical solution is known and by an example of a realistic flow through a complex arterial system.

A conservative, weakly nonlinear semi-implicit finite volume scheme for the compressible Navier - Stokes equations with general equation of state

A family of numerical models, known as the TRIM models, shares the same modeling philosophy for solving the shallow water equations. A characteristic analysis of the shallow water equations points out that the numerical instability is controlled by the gravity wave terms in the momentum equations and by the transport terms in the continuity equation. A semiimplicit finite-difference scheme has been formulated so that these terms and the vertical diffusion terms are treated implicitly and the remaining terms explicitly to control the numerical stability and the computations are carried out over a uniform finite-difference computational mesh without invoking horizontal or vertical coordinate transformations. An unstructured grid version of TRIM model is introduced, or UnTRIM (pronounces as “you trim”), which preserves these basic numerical properties and modeling philosophy, only the computations are carried out over an unstructured orthogonal grid. The unstructured grid offers the flexibilities in representing complex study areas so that fine grid resolution can be placed in regions of interest, and coarse grids are used to cover the remaining domain. Thus, the computational efforts are concentrated in areas of importance, and an overall computational saving can be achieved because the total number of grid-points is dramatically reduced. To use this modeling approach, an unstructured grid mesh must be generated to properly reflect the properties of the domain of the investigation. The new modeling flexibility in grid structure is accompanied by new challenges associated with issues of grid generation. To take full advantage of this new model flexibility, the model grid generation should be guided by insights into the physics of the problems; and the insights needed may require a higher degree of modeling skill.