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Session Overview
MS 21: High order schemes for time dependent problems: embedded boundaries, block finite differences, fluid dynamics, convergence analysis
Thursday, 15/July/2021:
12:00pm - 2:00pm

Session Chair: Jean-Pierre Croisille
Session Chair: Adi Ditkowski
Session Chair: Dalia Fishelov
Virtual location: Zoom 4

Session Abstract

High order approximation methods for complex physical models remain an essential objective

in applied mathematics. The purpose of this minisymposium is to provide an overview on

recent developments on this topic.

The emphasis will be on high order finite difference schemes for time dependent problems.

Also, a particular attention will be devoted to recent developments in finite difference theory:

embedded boundaries methods in Cartesian grids, methods to impose boundary conditions

and/or measurements, relations to high order finite elements, time stepping schemes. The

mathematical analysis will also be emphasized: matrix methods, SAT and SBP techniques,

etc. Specific applications including fluid and wave dynamics are welcome. Studies on

relations between various high order methods (FV, DG, SE), and their analysis is also


The targeted audience consists both of senior experts and also of young researchers and post

docs with a mathematical, numerical or physical background. The general goal is to attract

attention of the audience on high order methods as a modern branch of applied mathematics,

which it was historically.

Particular topics of interest are

* pact schemes for fluid dynamics problems

* embedded boundary techniques

* block finite difference schemes

* frameworks for the mathematical convergence analysis of high order methods for time dependent problems.

* relation of compact schemes with high order methods: Finite Elements, Finite Volumes, Spectral Elements, ...

* numerical experiments for high order methods: coupling, penalisation, etc.

* new ideas for the mathematical analysis of physical phenomena emerging from numerical experiments.

This list of topics is not exclusive. Overall a main objective is to insist on the intertwining

between the continuous approach and high order discrete methods and how the latter better

represent the complexity of the continuous problems.

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12:00pm - 12:30pm

A Fourth Order Compact Scheme with Difference Potentials for the 3D Wave Equation

Fouche Smith1, Semyon V Tsynkov1, Eli Turkel2

1North Carolina State University, United States of America; 2Tel Aviv University, Israel

We construct a compact fourth order accurate finite difference scheme, in both space and time, for the three-dimensional wave (d’Alembert) equation. The wave equation may have either constant or variable propagation speed. The scheme is built of the Cartesian grid. It is implicit in time and uses a 3x3x3 stencil in space on three consecutive time levels. A compact stencil means that the scheme requires no auxiliary boundary conditions beyond those needed by the underlying differential equa-tion itself. The scheme is implemented by forming a right-hand side composed of the solution on the previous two time levels and then solving an elliptic equation to obtain the solution on the upper time level. In the case of a wave equation, this elliptic equation reduces to the negative definite mod-ified Helmholtz equation.

The inversion at the upper time level is done by multigrid, which demonstrates fast convergence. If we take the solution from the previous time level as the initial guess, only a small number of multi-grid cycles proves sufficient for driving the error of the iterative solution down to the level of the discretization error on the grid. The overall complexity remains comparable to that of an explicit method. Altogether, the scheme is considerably more efficient than its lower order counterparts even though it is implicit and only conditionally stable.

To handle curvilinear boundaries that do not conform to the Cartesian discretization grid, the scheme is implemented along with the method of difference potentials (MDP). The MDP is applied to the modified Helmholtz equation on the upper time level. It enables numerical solution with no deterioration of accuracy (fourth order) regardless of the shape of the boundary. Moreover, it shows no adverse effect on the stability of time marching due to the cut cells. Finally, the MDP of-fers a straightforward and universal treatment of a broad variety of boundary conditions (Dirichlet, Neumann, Robin, etc.).

In the case of an exterior problem, the computational domain is terminated by the Bayliss-Turkel (BT) radiation boundary conditions. Higher order BT conditions are implemented using the auxilia-ry variables. An alternative treatment of the artificial outer boundary involves a PML approximated on the grid with fourth order accuracy, to match the accuracy of the interior approximation.

Computational examples demonstrating the efficiency of the proposed scheme will be presented.

Work supported by the US Army Research Office (ARO) under grant number W911NF-16-1-0115 and by the US--Israel Binational Science Foundation (BSF) under grant number 2014048.

12:30pm - 1:00pm

Block Finite Difference Schemes and Their Relations to Finite Element methods.

Adi Ditkowski1, Anne Le Blanc2, Chi-Wang Shu3

1Tel Aviv Univerrsity, Israel; 2Tel Aviv Univerrsity, Israel; 3Brown University, USA

Block Finite Difference schemes (BFD) are difference-methods in which the domain is divided into blocks, or cells, containing two or more grid points with a different scheme used for each grid point, unlike the standard FD method. BFD methods have a similar algebraic structure to nodal-based Finite Element methods, in particular, DG method. We show that BFD schemes can indeed be presented as particular kinds of nodal-based FE schemes.

In some cases, these schemes can be constructed such that the actual errors

are much smaller than their truncation errors. This error reduction is made by

constructing the schemes such that they inhibit the accumulation of the local errors. Therefore, they are called Error Inhibiting Schemes (EIS).

This equivalency between BFD and FE methods allows us to derive highly ac-

curate FE schemes. For example, this FE scheme with p = 2 can be fourth or

even fifth-order schemes. In some cases, even a sixth-order convergence rate can be obtained using a post-processing filter.

1:00pm - 1:30pm

Flexible semi-bounded schemes for large geometry deformations using novel formulations of ALE and AMR

Tomas Lundquist1, Andrew Ross Winters1, Jan Nordström1, Arnaud George Malan2

1Linköping University, Sweden; 2University of Cape Town, South Africa

By introducing a moving frame of reference, Arbitrary Lagrangian Eulerian (ALE) schemes are designed to accommodate the continuous movement of geometric boundaries during the course of a flow simulation. In the traditional sense, ALE methods are seen as specialized techniques relying on the careful interplay between spatial and temporal approximations as well as the mesh motion itself. By contrast, time dependent Adaptive Mesh Refinement (AMR) is ideal for reducing the cost of a simulation by adding or removing nodes where needed based on changing local flow characteristics. However, AMR must sometimes also be applied in conjunction with ALE in order to avoid deterioration of mesh quality associated with large geometric movements. When used together, both the ALE scheme and AMR interpolation can be applied locally inside a limited region in space close to the moving boundary, while leaving the mesh stationary in the rest of the domain. In this talk we will discuss and relate two recent papers on ALE and AMR techniques respectively, both from the perspective of semi-bounded spatial discretizations, energy stability and the method-of-lines. By using novel formulations, we will combine spatial, temporal and AMR interpolation techniques in a building-block block manner such that the energy stability of the equivalent problem on a fixed mesh is automatically preserved. In particular, arbitrary semi-bounded spatial discretizations can be applied together with standard explicit or implicit time stepping schemes, with similar properties of stability and free-stream preservation as for the stationary mesh case. In this way, arbitrary movements of geometric boundaries can be accommodated in a flexible, simple and effective way with minimal changes to existing methods.

1:30pm - 2:00pm

Time-Dependent Fourth-Order Problems: Error Analysis

Dalia Fishelov1, Jean-Pierre Croisille2, Matania Ben-Artzi3

1Afeka Tel Aviv Academic College of Engineering, Israel; 2Univ. de Lorraine, Metz 57045, France; 3The Hebrew University, Jerusalem 91904, Israel

The two-dimensional incompressible Navier-Stokes (NS) equations

$$ (\Delta\psi)_t+(\nabla^{\perp}\psi) \cdot\nabla(\Delta\psi) =\nu\Delta^2 \psi,$$

where $\psi$ is the streamfunction, play an important role in various areas of physics. In our book [1] we have suggested fourth-order compact schemes for the Navier-Stokes equations. The same type of schemes may be applied to the clamped plate problem $\partial_t u +\Delta^2 u =f,$ which serves as a linear model to the 2D Navier-Stokes equations. Another problem which we handle contains all the linear components of the Navier-Stokes equations in stream-function formulation $$\partial_t\Delta u=\Delta^2 u+f$$.

The numerical methods suggested in [1] for the Navier-Stokes equation

have truncation errors which are of fourth-order O(h^4) at interior points,

but are only of first-order O(h) at near-boundary points,

We have proved in [1] an optimal convergence theorem, that provides a bound on the error of our high-order compact scheme for $u_{xxxx}=f$, which is O(h^4).

In addition, we have shown that, for the semi-discretization of time-dependent equation $u_t+u_{xxxx}=f$, the error is bounded by $Ch^{4-\epsilon}$, where $\epsilon$ arbitrary small (see [2]).

A similar result hold for the equation $u_t+u_{xxxx}+d~u_xx+ a~u+f$, for large enough $a$.

The situation, where the scheme has lower order truncation error at near boundary points compared to high-order truncation errors at interior points, occurs also for the high-order finite difference schemes suggested in [3] and [4].

It was proved in[3] and [4] that for the error in the solution decays to zero at the same rate of the decay to zero of the truncation error at interior points.

We prove that for high-order compact schemes, which were presented in [1],

a similar theorem may be proved.

The new approach in our proofs is based on vector representation of the error and on matrix representation of the spatial operators. Therefore, the error in the solution, which is represented as a vector, satisfies an ordinary differential equation in vector form. The above system may be solved for linear problems. Then, the optimal convergence theorem (see [1]) is invoked, together with the extended mean-value theorem for integration.

In particular, for the clamped plate problem $\partial_t u +\Delta^2 u =f$,

the error tends to zero as $O(h^4)$. A similar result result holds for the one-dimensional problem $\partial_t\Delta u=\Delta^2 u+f$.


[1] M. Ben-Artzi, J.-P. Croisille and D. Fishelov, ``Navier-Stokes Equations in Planar

Domains'', Imperial College Press, 2013.

[2] M. Ben-Artzi, J.-P. Croisille and D. Fishelov, "Time evolution of discrete

fourth- order elliptic operators", Numerical Analysis Partial Differential Equations

(NMPDE), Vol. 35 (2019), pp. 1429-1457.

[3] B. Gustafsson, "The convergence rate for difference approximations to general

mixed initial boundary value problems", SIAM J. Numer. Anal., Vol. 18 (1981),

pp. 179-190.

[4] S. Abarbanel, A. Ditkowski and B. Gustafsson, "On Error Bounds of Finite

Difference Approximations to Partial Differential Equations Temporal Behavior

and Rate of Convergence", J. Sci. Comput., Vol.f 15 (2000), pp. 79-116.

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