使用勒让德-高斯-洛巴托（LGL）配置求解边值问题（BVP），恒星：2，更新：2024-06-22 1213.rar

## Pseudospectral methods for boundary value problems Given below is a boundary value problem from Ref. [^1]. $$ \begin{equation} \ddot{x}-t\dot{x}-x=0 \end{equation} $$ $$ \begin{aligned} x(t=-1)&=1\\ x(t=1)&=1 \end{aligned} $$ $x^*(t)=\exp(\frac{t^2-1}{2})$ is the solution to the boundary value problem. ### Numerical solution Pseudospectral collocation requires the following steps 1. discretize domain, state and control at $n+1$ points 2. discrete state and control input are the variables 3. approximate state $(x(t))$ and control $(u(t))$ using global polynomials 4. differentiate the polynomial to the required degree and evaluate at the collocation points 5. evaluate the boundary value problem at the collocation points * dynamics evaluated at interior points * boundary conditions at -1 and 1 8. depending on the dynamics, system of linear/nonlinear system of equations need to be solved ## Implementation An optimization problem without an objective function and inequality constraints is simply a system of equation which may be linear or nonlinear. Using the principle, we use IPOPT to solve the system of nonlinear equations. CasADi's ```Opti()``` (meant as modelling language for optimization problems) is used to model the equations. The codes serve the following purposes. * ```bvp_main.m``` - solve the BVP for a given grid size and compare with analytical solution * ```bvp.m, convergence.m``` - study exponential/spectral convergence of discretization error * ```legslb.m, legslbdiff.m, lepoly.m, lepolym.m``` - supplementary files for LGL collocation ## Requirements * OCTAVE-6.1.0 * CasADi-3.5.5 ## Comparison of numerical and analytical results ### side by side plot  ### error plot  ### Spectral/exponential convergence A key feature of pseudospectral collocation is rapid decrease in discretization error with increasing grid size. However, for larger N ($\geq 100$), the problem can get numerically ill-conditioned. Some methods to mititgate ill-conditioning include using knotting, preconditioners and Birkhoff collocation.  ## References [^1]: Wang, L. L., Samson, M. D., & Zhao, X. (2014). A well-conditioned collocation method using a pseudospectral integration matrix. SIAM Journal on Scientific Computing, 36(3), A907-A929.