High Accuracy Approximations of Burgers’ Equation Using Adjoint Data Assimilation Methods

Progress in modeling complex phenomena in many areas of physics, including fluid mechanics, electromagnetism, quantum mechanics, general relativity, and materials science requires solving non-linear Partial Differential Equations (PDEs).Burgers' equation is especially important, as it is one of the simplest non-linear equations in physics that captures essential features of more complex physical phenomena.In this work Burgers' equation is solved both exactly and numerically with the following as initial conditions: the sinusoidal function, Dirac delta function, and Heaviside step function.For the numerical solution, I use a predictor-correct scheme through time, and a finite difference scheme with 4th-order accuracy through space.Numerical results from both low viscosity and high viscosity conditions are compared with exact solutions.Before Data Assimilation (DA) is applied, even with decent numerical accuracy, the difference between the numerical result and exact solution is visible at the end of the analysis time window, which occurs due to the accumulation of numerical error while integrating.I then derive and apply the DA algorithm based on the adjoint method for Burgers' equation.The DA-based solution from the adjoint equation successfully corrects the initial condition and significantly improves the numerical results.The DA approach outlined in this research not only works for approximations of Burgers' equation, but it also has applications to other non-linear partial differential equations.