111 research outputs found
ADER-WENO Finite Volume Schemes with Space-Time Adaptive Mesh Refinement
We present the first high order one-step ADER-WENO finite volume scheme with
Adaptive Mesh Refinement (AMR) in multiple space dimensions. High order spatial
accuracy is obtained through a WENO reconstruction, while a high order one-step
time discretization is achieved using a local space-time discontinuous Galerkin
predictor method. Due to the one-step nature of the underlying scheme, the
resulting algorithm is particularly well suited for an AMR strategy on
space-time adaptive meshes, i.e.with time-accurate local time stepping. The AMR
property has been implemented 'cell-by-cell', with a standard tree-type
algorithm, while the scheme has been parallelized via the Message Passing
Interface (MPI) paradigm. The new scheme has been tested over a wide range of
examples for nonlinear systems of hyperbolic conservation laws, including the
classical Euler equations of compressible gas dynamics and the equations of
magnetohydrodynamics (MHD). High order in space and time have been confirmed
via a numerical convergence study and a detailed analysis of the computational
speed-up with respect to highly refined uniform meshes is also presented. We
also show test problems where the presented high order AMR scheme behaves
clearly better than traditional second order AMR methods. The proposed scheme
that combines for the first time high order ADER methods with space--time
adaptive grids in two and three space dimensions is likely to become a useful
tool in several fields of computational physics, applied mathematics and
mechanics.Comment: With updated bibliography informatio
Numerical methods for advection-diffusion-reaction equations and medical applications
The purpose of this thesis is twofold, firstly, the study of a relaxation procedure for numerically solving advection-diffusion-reaction equations, and secondly, a medical application. Concerning the first topic, we extend the applicability of the Cattaneo relaxation approach to reformulate time-dependent advection-diffusion-reaction equations, that may include stiff reactive terms, as hyperbolic balance laws with stiff source terms. The resulting systems of hyperbolic balance laws are solved by extending the applicability of existing high-order ADER schemes, including well-balanced and non-conservative schemes. Moreover, we also present a new locally implicit version of the ADER method to solve general hyperbolic balance laws with stiff source terms. The relaxation procedure depends on the choice of a relaxation parameter . Here we propose a criterion for selecting in an optimal manner, relating the order of accuracy of the numerical scheme used, the mesh size and the chosen . This results in considerably more efficient schemes than some methods with the parabolic restriction reported in the current literature. The resulting present methodology is validated by applying it to a blood flow model for a network of viscoelastic vessels, for which experimental and numerical results are available. Convergence-rates assessment for some selected second-order model equations, is carried out, which also validates the applicability of the criterion to choose the relaxation parameter. The second topic of this thesis concerns the numerical study of the haemodynamics impact of stenoses in the internal jugular veins. This is motivated by the recent discovery of a range of extra cranial venous anomalies, termed Chronic CerbroSpinal Venous Insufficiency (CCSVI) syndrome, and its potential link to neurodegenerative diseases, such as Multiple Sclerosis. The study considers patient specific anatomical configurations obtained from MRI data. Computational results are compared with measured data
Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes
We present a new family of high order accurate fully discrete one-step
Discontinuous Galerkin (DG) finite element schemes on moving unstructured
meshes for the solution of nonlinear hyperbolic PDE in multiple space
dimensions, which may also include parabolic terms in order to model
dissipative transport processes. High order piecewise polynomials are adopted
to represent the discrete solution at each time level and within each spatial
control volume of the computational grid, while high order of accuracy in time
is achieved by the ADER approach. In our algorithm the spatial mesh
configuration can be defined in two different ways: either by an isoparametric
approach that generates curved control volumes, or by a piecewise linear
decomposition of each spatial control volume into simplex sub-elements. Our
numerical method belongs to the category of direct
Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation
formulation of the governing PDE system is considered and which already takes
into account the new grid geometry directly during the computation of the
numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a
posteriori sub-cell finite volume limiter method, in which the validity of the
candidate solution produced in each cell by an unlimited ADER-DG scheme is
verified against a set of physical and numerical detection criteria. Those
cells which do not satisfy all of the above criteria are flagged as troubled
cells and are recomputed with a second order TVD finite volume scheme. The
numerical convergence rates of the new ALE ADER-DG schemes are studied up to
fourth order in space and time and several test problems are simulated.
Finally, an application inspired by Inertial Confinement Fusion (ICF) type
flows is considered by solving the Euler equations and the PDE of viscous and
resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure
The Montecinos-Balsara ADER-FV polynomial basis: Convergence properties & extension to non-conservative multidimensional systems
Hyperbolic systems of PDEs can be solved to arbitrary orders of accuracy by
using the ADER Finite Volume method. These PDE systems may be non-conservative
and non-homogeneous, and contain stiff source terms. ADER-FV requires a
spatio-temporal polynomial reconstruction of the data in each spacetime cell,
at each time step. This reconstruction is obtained as the root of a nonlinear
system, resulting from the use of a Galerkin method. It was proved in Jackson
[7] that for traditional choices of basis polynomials, the eigenvalues of
certain matrices appearing in these nonlinear systems are always 0, regardless
of the number of spatial dimensions of the PDEs or the chosen order of accuracy
of the ADER-FV method. This guarantees fast convergence to the Galerkin root
for certain classes of PDEs.
In Montecinos and Balsara [9] a new, more efficient class of basis
polynomials for the one-dimensional ADER-FV method was presented. This new
class of basis polynomials, originally presented for conservative systems, is
extended to multidimensional, non-conservative systems here, and the
corresponding property regarding the eigenvalues of the Galerkin matrices is
proved
- …