Short communicationThe exact formula of the optimal penalty parameter value of the spectral penalty method for differential equations
Introduction
Spectral methods are still popular in solving differential equations in various application areas. Spectral methods provide highly accurate solutions to differential equations when solutions are smooth. If the solution is smooth enough, the spectral approximation converges exponentially fast. Spectral methods as a global method are known to be superior to local methods such as finite difference methods although for non-smooth problems finite difference approach such as compact scheme may yield more accurate results than spectral method [1]. For its high accuracy spectral methods are also used recently for fractional differential equations [9].
Stability and accuracy of spectral solutions can be improved further by imposing the given boundary conditions weakly, i.e. by penalizing the boundary conditions. Here we note that a slight different approach is also possible such as the one found in Driscoll and Hale [2] instead of removing or weakening the boundary conditions in the spectral formulation. The spectral penalty method was originally proposed in Funaro [3] for solving elliptic problems and later for hyperbolic problems in Funaro and Gottlieb [4]. The flexibility of the penalty method makes the spectral method suitable for dealing with boundary conditions in irregular domains. For this reason, it is also known that there are close relations between discontinuous Galerkin methods and spectral methods with multi-domains and elements. Extensive works on penalty methods have been made (for more discussions see [6]). There have been rigorous studies on the stability of the spectral penalty method [5]. Penalty methods can be also used for fractional differential equations [8]. The study on the optimization of its accuracy rather relies on numerical experiments, e.g. [7]. In this short note, we consider a simple differential equation and derive the optimal penalty parameter value that minimizes the maximum error. We verify that the derived formula matches the expected results well numerically. We note that our exact formula is valid only for ODEs of the type in Eq. (2.1).
Section snippets
Spectral penalty method
Consider the following first order differential equation for u ∈ C1with the boundary condition and appropriate f(x). We are interested in solving Eq. (2.1) using the Chebyshev spectral penalty collocation method based on the Chebyshev polynomials of Tn(x). Using other polynomials rather than the Chebyshev polynomials may yield different exact form. With the penalty method, the boundary condition is embedded in the new penalized equation. The numerical solution with
Numerical verification
For the numerical verification, consider the following differential equationwith the boundary condition . The exact solution is . We seek the numerical solution of degree at most to verify the optimal value of τ derived in the previous section. The numerical solution could be sought in PN, for which a small modification from to N can be used to adjust the formula, Eq. (2.13). In Figs. 1 and 2 the numerical
Conclusion
In this note, we considered the numerical solution of a simple first order differential equation with the Chebyshev spectral penalty method. With the penalty method, the boundary condition is penalized. In this note we showed that the maximum absolute error exists at either the first or the second collocation point. Based on such an observation we calculated the optimal penalty parameter value, that minimizes the maximum error, and verified the obtained value numerically. The absolute optimal
Acknowledgment
This work has been partially supported by Ajou University. The authors thank the anonymous reviewers for their helpful comments.
References (9)
- et al.
Spectral Methods for Time-Dependent Problems
(2007) - et al.
Exponentially accurate spectral and spectral element methods for fractional ODEs
J. Comput. Phys.
(2014) - et al.
On the comparison between compact finite difference and pseudospectral approaches for solving similarity boundary layer problems
Math. Probl. Eng.
(2013) - et al.
Rectangular spectral collocation
IMA J. Numer. Anal.
(2015)