Adaptive High-Order Methods for Fluid Flows

Authors: Lott & Deane

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Applied Mathematics and Scientific Computation

The goal of this research is to construct an Adaptive Spectral Element Method to solve non-linear PDEs.

We aim to create an efficient method for solving large scale PDE's on complex domains. We use adaptivity to maintain a high working accuracy for long time periods without wasting computational resources. Spectral Element

Methods are computationally efficient for higher dimensional problems due to their tensor product formulation of the system operators.

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The Spectral Element Method approximates the solution to the PDE by

partitioning the domain into elements, and on each element, solving an

integral equation. In order to solve the equation, one constructs a

weighted sum of polynomial basis functions. In particular, SEM uses the

Gauss-Lobatto-Legendre quadrature rules in order to obtain an exact

solution for integrands of order 2P+1 or less, when using Legendre

polynomials of degree P. This gives SEM an exponential convergence

property that methods such as Finite Element, or Finite Difference

methods dont have. As we increase the polynomial degree in the SEM,

we exponentially approach the solution to the integral equation. In low

order methods, where only the number of elements is increased, one

obtains only algebraic convergence to the solution.

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Solution to 1D Viscous Burgers Equation using P-Adaptivity

We have implemented error estimators that

determine when to increase or decrease the

polynomial degree on each element in order to

obtain an accurate solution without wasting

computational resources in well behaved areas

of the solution.

To the right, we show the computational work

required to integrate a linear advection equation

Slope of solution at x=0, within 5 periods while maintaining a phase shift error

99.85% of the analytical value at <.1 (Karniadakis).
max slope.
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Gauss Legendre Lobatto Nodes
4 Elements Polynomial Degree 5
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After discretizing in space and time, we are left with a matrix equation. In
1D, the matrices are block diagonal, with each block corresponding to a
single element. The block size is (p+1)^2, where p is the polynomial
degree of the approximation on that element. In higher dimensions, the
system matrices are formed by applying kronecker tensor products of the
1D operators. Thus
There is very little storage overhead for the higher dimensional systems.
Moreover, the kronecker tensor formulation reduces the order of
operations from O(n^(d+2)) to O(n^(d+1)) for a d-dimensional
calculation with n grid points.
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Element 2
Global Ordering
Element 1
Element 2
Local Ordering
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Linear advection of a 2D Gaussian wave for one time peroid.
Uncoupled Stiffness Matrix
(Laplacian Operator)
Element 1
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We determine a global ordering of each node of the discretization, this
ordering determines the structure of the coupled system operators.
However, because we are using iterative solvers, we never need to
construct the global system. Instead, we compute the contribution of each
element to the solution, and then use a weighted sum to obtain the global
solution, taking into account where the same node is used on two
elements. For example the global solution at node 7 in the global ordering
is obtained by adding the solution at node 7 from element 1 and node 1
from element 2, and then dividing by 2. The name of this weighted
summation operation is referred to as Direct-Stiffness-Summation (DSS).
In order to parallelize the SEM, each processor is assigned a group of
elements whose solution it will compute. Each time a solution on those
elements is computed, a parallel DSS is used to determine the nodal
values at (non-local) element Boundaries. For example if element 1 and
element 2 are on separate processors, each time a solution is computed
on both of them, the solution at node 7 on processor 1 is summed with the
solution at node 1 on processor 2. The result is then divided by two and
stored on both processors as the value of the solution at that node. In
order to determine the dependencies between processors for complicated
geometries, a parallel bin sort is used. This past summer we implemented
this parallel DSS technique using MPI.
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We have implemented 2D Advection Diffusion Equation, including non-linear advection.
This code allows for periodic, and Dirichlet Boundary conditions. We have designed our
code in an object oriented fashion to allow us to add new capabilities. Future Research
plans include implementing the Navier-Stokes Equations, curved elements, adaptivity in
higher dimensions, and Preconditioned Iterative Schemes.
References
[1] Anil Deane. Spectral and spectral-element methods: Lecture notes in high performance computational physics.
NASA Contractor Report 203877, 1997.
[2] P.F. Fischer. An overlapping schwarz method for spectral element solution of the incompressible navier-stokes
equations. Journal of Computational Physics, 1997
[3] S.J. Sherwin G.E. Karniadakis. Spectral/hp Element Methods for CFD. Numerical Mathematics and Scientific
Computation. Oxford University Press, Oxford, 1999.
[4] Rainald Lhner. An adaptive finite element scheme for transient problems in cfd. Computational Methods in
Applied Mechanics and Engineering, 61:323338, 1987.
[5] P. Aaron Lott. Project website. http://www.lcv.umd.edu/ ~palott/research/graduate/663/.
[6] E.H. Mund M.O. Deville, P.F. Fischer. High-Order Methods for Incompressible Fluid Flows. Cambridge Monographs
on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2002.