342 research outputs found
Optimized explicit Runge-Kutta schemes for the spectral difference method applied to wave propagation problems
Explicit Runge-Kutta schemes with large stable step sizes are developed for
integration of high order spectral difference spatial discretization on
quadrilateral grids. The new schemes permit an effective time step that is
substantially larger than the maximum admissible time step of standard explicit
Runge-Kutta schemes available in literature. Furthermore, they have a small
principal error norm and admit a low-storage implementation. The advantages of
the new schemes are demonstrated through application to the Euler equations and
the linearized Euler equations.Comment: 37 pages, 3 pages of appendi
On improving the efficiency of ADER methods
The (modern) arbitrary derivative (ADER) approach is a popular technique for
the numerical solution of differential problems based on iteratively solving an
implicit discretization of their weak formulation. In this work, focusing on an
ODE context, we investigate several strategies to improve this approach. Our
initial emphasis is on the order of accuracy of the method in connection with
the polynomial discretization of the weak formulation. We demonstrate that
precise choices lead to higher-order convergences in comparison to the existing
literature. Then, we put ADER methods into a Deferred Correction (DeC)
formalism. This allows to determine the optimal number of iterations, which is
equal to the formal order of accuracy of the method, and to introduce efficient
-adaptive modifications. These are defined by matching the order of accuracy
achieved and the degree of the polynomial reconstruction at each iteration. We
provide analytical and numerical results, including the stability analysis of
the new modified methods, the investigation of the computational efficiency, an
application to adaptivity and an application to hyperbolic PDEs with a Spectral
Difference (SD) space discretization
Systematic construction of efficient six-stage fifth-order explicit Runge-Kutta embedded pairs without standard simplifying assumptions
This thesis examines methodologies and software to construct explicit
Runge-Kutta (ERK) pairs for solving initial value problems (IVPs) by
constructing efficient six-stage fifth-order ERK pairs without
standard simplifying assumptions. The problem of whether efficient
higher-order ERK pairs can be constructed algebraically without the
standard simplifying assumptions dates back to at least the 1960s,
with Cassity's complete solution of the six-stage fifth-order order
conditions. Although RK methods based on the six-stage fifth-order
order conditions have been widely studied and have continuing
practical importance, prior to this thesis, the aforementioned
complete solution to these order conditions has no published usage
beyond the original series of publications by Cassity in the 1960s.
The complete solution of six-stage fifth-order ERK order conditions
published by Cassity in 1969 is not in a formulation that can easily
be used for practical purposes, such as a software implementation.
However, it is shown in this thesis that when the order conditions are
solved and formulated appropriately using a computer algebra system
(CAS), the generated code can be used for practical purposes and the
complete solution is readily extended to ERK pairs. The condensed
matrix form of the order conditions introduced by Cassity in 1969 is
shown to be an ideal methodology, which probably has wider
applicability, for solving order conditions using a CAS. The software
package OCSage developed for this thesis, in order to solve the order
conditions and study the properties of the resulting methods, is built
on top of the Sage CAS.
However, in order to effectively determine that the constructed ERK
pairs without standard simplifying assumptions are in fact efficient
by some well-defined criteria, the process of selecting the
coefficients of ERK pairs is re-examined in conjunction with a
sufficient amount of performance data. The pythODE software package
developed for this thesis is used to generate a large amount of
performance data from a large selection of candidate ERK pairs found
using OCSage. In particular, it is shown that there is unlikely to be
a well-defined methodology for selecting optimal pairs for
general-purpose use, other than avoiding poor choices of certain
properties and ensuring the error coefficients are as small as
possible. However, for IVPs from celestial mechanics, there are
obvious optimal pairs that have specific values of a small subset of
the principal error coefficients (PECs). Statements seen in the
literature that the best that can be done is treating all PECs equally
do not necessarily apply to at least some broad classes of IVPs. By
choosing ERK pairs based on specific values of individual PECs, not
only are ERK pairs that are 20-30% more efficient than comparable
published pairs found for test sets of IVPs from celestial mechanics,
but the variation in performance between the best and worst ERK pairs
that otherwise would seem to have similar properties is reduced from a
factor of 2 down to as low as 15%. Based on observations of the small
number of IVPs of other classes in common IVP test sets, there are
other classes of IVPs that have different optimal values of the PECs.
A more general contribution of this thesis is that it specifically
demonstrates how specialized software tools and a larger amount of
performance data than is typical can support novel empirical insights
into numerical methods
Improvement of the Prediction of Surface Ozone Concentration Over Conterminous U.S. by a Computationally Efficient Second-Order Rosenbrock Solver in CAM4-Chem
The global chemistry-climate model (CAM4-Chem) overestimates the surface ozone concentration over the conterminous U.S. (CONUS). Reasons for this positive bias include emission, meteorology, chemical mechanism, and solver. In this study, we explore the last possibility by examining the sensitivity to the numerical methods for solving the chemistry equations. A second-order Rosenbrock (ROS-2) solver is implemented in CAM4-Chem to examine its influence on the surface ozone concentration and the computational performance of the chemistry program. Results show that under the same time step size (1800 s), statistically significant reduction of positive bias is achieved by the ROS-2 solver. The improvement is as large as 5.2 ppb in Eastern U.S. during summer season. The ROS-2 solver is shown to reduce the positive bias in Europe and Asia as well, indicating the lower surface ozone concentration over the CONUS predicted by the ROS-2 solver is not a trade-off consequence with increasing the ozone concentration at other global regions. In addition, by refining the time step size to 180 s, the first-order implicit solver does not provide statistically significant improvement of surface ozone concentration. It reveals that the better prediction from the ROS-2 solver is not only due to its accuracy but also due to its suitability for stiff chemistry equations. As an added benefit, the computation cost of the ROS-2 solver is almost half of first-order implicit solver. The improved computational efficiency of the ROS-2 solver is due to the reuse of the Jacobian matrix and lower upper (LU) factorization during its multistage calculation
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