Spectral methods for stiff problems MATH 6646 JULIAN BREW REDA MOUDDEN

outline Presentation of the MEMS Device Presentation of the Spectral Collocation Methods Study of Bernstein polynomials Comparisons

I. Presentation of the MEMS Device Doubly supported beams with axial stress Used as pressure sensors : usually piezo-resistive or piezo-electric transducers that generate a signal as a function of the pressure imposed. Useful device in weather instrumentation, aircrafts, automobiles and any type of leak testing. w : Displacement Stiff ODE : E : Young modulus I : Moment of inertia q : Distributed load N : Axial force

II. Presentation of the spectral collocation methods Spectral Methods Spectral methods are a class of techniques that numerically solve differential equations. The idea is to write the solution of the differential equation as a sum of certain basis functions and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible. Characteristics of these methods include : Global method Continuous solutions Depends on several parameters Difficult to combine with sharp gradients Algorithm and CPU strong dependence Several parameters : Order of the highest polynomial, collocation points, time length

Theory II. Presentation of the spectral collocation methods Assume solution is weighted sum of basis functions 𝑦 (𝑥)= 𝑖=0 𝑚 𝑐 𝑖 𝜙 𝑖 (𝑥) Substitute assumed solution into original ODE to get a constraint equation 𝑦 ′ 𝑥 = 𝑖=0 𝑚 𝑐 𝑖 𝑑 𝜙 𝑖 𝑑𝑥 𝑥 =𝑓 𝑥, 𝑦 𝑥 At each node 𝑥 𝑗 , the ODE constraint should be satisfied 𝑦 ′ 𝑥 𝑗 = 𝑖=0 𝑚 𝑐 𝑖 𝑑 𝜙 𝑖 𝑑𝑥 𝑥 𝑗 =𝑓 𝑥 𝑗 , 𝑦 𝑥 𝑗 Plugging in each node 𝑥 𝑗 and boundary condition into the constraint equation, a system of equations in terms of 𝑐 𝑖 is found 𝐹 𝑐 = 0 Formulation generalizes to 𝑦 ∈ ℝ 𝑛 and to BVP ODE: 𝑦 ′ =𝑓 𝑥,𝑦 with boundary conditions on 𝑦 or 𝑦’

Example II. Presentation of the spectral collocation methods y’’ – y = x y(0) = 0 y(1) = 0 x ∈[0;1] We choose 2 collocation points x = 0.25 and x = 0.5 We get 4 points total (2 from the boundary conditions and 2 from the collocation) => 3rd order polynomial : 𝑦 = 𝛼 0 + 𝛼 1 .𝑥+ 𝛼 2 . 𝑥 2 + 𝛼 3 . 𝑥 3 𝑦 ′= 𝛼 1 + 2𝛼 2 .𝑥+3 𝛼 3 . 𝑥 2 𝑦 ′′= 2𝛼 2 +6 𝛼 3 .𝑥 We input these functions in the ODE : − 𝛼 0 − 𝛼 1 .𝑥+ 2− 𝑥 2 . 𝛼 2 +(6𝑥− 𝑥 3 ). 𝛼 3 = 𝑥 Evaluating this equation at the 2 collocation points gives us 2 equations. − 𝛼 0 − 𝛼 1 (0.25)+ 2− 0.25 2 𝛼 2 +(6(0.25)− 0.25 3 ). 𝛼 3 = 0.25 − 𝛼 0 − 𝛼 1 (0.5)+ 2− 0.5 2 𝛼 2 +(6(0.5)− 0.5 3 ). 𝛼 3 = 0.5 Using the boundary conditions, we get the other 2 equations. 𝛼 0 + 𝛼 1 0 + 𝛼 2 0 2 + 𝛼 3 0 3 = 𝛼 0 =0 𝛼 0 + 𝛼 1 1 + 𝛼 2 1 2 + 𝛼 3 1 3 = 𝛼 0 + 𝛼 1 + 𝛼 2 + 𝛼 3 =0 Numerical solution : 𝑦 𝑥 =−0.1459𝑥 −0.006738 𝑥 2 +0.1526 𝑥 3

II. Presentation of the spectral collocation methods Collocation points The collocation points are a set of nodes at which the polynomial basis equals the true solution. There are several collocation methods, the most commonly used are the following : For these polynomials, the concentration of nodes increases towards the edges of the interval. This results in oversampling and high-frequency oscillations at the end (Runge phenomenon).

Choice of a collocation point method II. Presentation of the spectral collocation methods Choice of a collocation point method Chebyshev vs Equally-spaced vs Legendre-Gauss-Lobato m = 5 CPU Time (ms) Max Error (E-11) Total Error (E-12) Equally Spaced 13 4.18 3.59 LQL 12 1.58 1.14 Chebyshev 7 1.33 0.64 m : highest polynomial order m = 3 CPU Time (ms) Max Error (E-10) Total Error (E-11) Equally Spaced 8 2.57 2.12 LQL 9 1.92 1.41 Chebyshev 5 1.94 0.9

Bernstein polynomials III. Study of Bernstein polynomials Bernstein polynomials The Bernstein polynomials of degree m are defined by In general, we approximate any function as follows Where and N.B. When compared to other basis polynomials, the Bernstein form is systematically more stable, when evaluating polynomials whose coefficient are subject to random perturbations.

Influence of the order of the polynomial III. Study of Bernstein polynomials Influence of the order of the polynomial m = 4 m = 5 m = 6 N.B. There exists a maximum m, after which, Solving system becomes numerically difficult Solution accuracy is bounded by machine precision

Potential issues III. Study of Bernstein polynomials Solving ODE using spectral collocation methods becomes difficult when solving the system of equations becomes numerically challenging These numerical issues may occur when Highest polynomial order, m, is large Time horizons are large Original ODE is stiff Multiple local solutions exist Mitigations to these issues Solve lower order basis polynomial solution and use to initialize higher order polynomial solution Center and scale coordinates to better condition system of equations Separate overall time horizon into smaller sub-problems Compute analytic Jacobian matrix Robust nonlinear system equation solver

IV. COMPARISONS Power series (1)

IV. COMPARISONS Power series (2)

IV. COMPARISONS RK45 (1)

IV. COMPARISONS RK45 (2)

conclusion Efficient way to solve stiff ODEs where RK45 may not given accurate solutions Use of Chebyshev collocation points : Minimization of Runge phenomenon, CPU time and error Use of Bernstein basis polynomials : Better numerical stability and CPU time MEMS device application Depending on the system, we may have numerical issues

References Philippe Grandclément, Introduction to spectral methods, (2006) M. Alshbool and I. Hashim, Bernstein polynomials for solving nonlinear stiff system of ordinary differential equations,060015,1–6 (2015) A. Akyüz-Dascıoglu and H. Çerdik-Yaslan, The solution of high-order nonlinear ordinary differential equations by Chebyshev Series, (2010) W. J. Thompson, Fourier series and the Gibbs phenomenon, (1991) Rida T. Farouki, The Bernstein polynomial basis: A centennial retrospective, Computer Aided Geometric Design, (2012) Brandon Jones, Orbit Propagation Using Gauss-Legendre Collocation, American Institute of Aeronautics and Astronautics/American Astronautical Society, (2012) Nairat Kanyamee and Zhimin Zhang, Comparison of a spectral collocation method and symplectic methods for Hamiltonian systems, International Journal of Numerical Analysis and Modeling, (2011)