Accuracy and Stability Analysis of Summation-By-Parts Operators


Summation-by-parts (SBP) operators are a type of finite-difference operator that is constructed with stability advantages. Their accuracy and stability properties are analyzed as the operators are implemented to solve the linear convection and the quasi-1D Euler equations in conjunction with the simultaneous-approximation-terms (SAT) boundary treatment. Accuracy is analyzed with a nodal root-mean-square error measurement, and stability is analyzed with the eigenspectrum method. Additionally, these two properties are compared to those of the centered-difference operators with conventional boundary schemes. The scalar and matrix dissipation models are also implemented, and their influence observed.

While the SBP schemes are less accurate than the conventional schemes in the Euler equations without dissipation, they become more accurate when dissipation is applied. Even though the eigenvalues from the SBP schemes appear to be more dissipative than those from the conventional schemes, they do not possess a clear stability advantage. The fourth-order SBP scheme has near-singular eigenvalues, and should be used with dissipation to assuage convergence difficulties; the fifth-order SBP scheme has eigenvalues with large imaginary parts, requiring very small Courant numbers for explicit time-marching methods.

X. Huan. Accuracy and Stability Analysis of Summation-By-Parts Operators. B.A.Sc. Thesis, University of Toronto, 2008.


author = {Huan, Xun},
school = {University of Toronto},
title = {{Accuracy and Stability Analysis of Summation-By-Parts Operators}},
type = {Bachelor's thesis},
year = {2008}