Computational Analysis of a duality based approach for solving ordinary and partial differential equations

Motivated by the need to solve physically well-founded, nonlinear partial differential equations (PDE) of non-standard form that arise in the modeling of defect dynamics in materials, this work focuses on the development of a novel computational approximation scheme for such problems, and demonstrates it on problems of lesser complexity with well-known prop erties. In the process, it also serves as an assessment of a variational principle based scheme for solving nonlinear PDE proposed in [1] and developed further in [2, 3]. A variety of criti cally chosen equations, which includes both initial and boundary value problems (IVPs and BVPs) and are referred to as the primal equations in this work, are transformed into (space) time BVPs, referred to as the dual equations. The dual BVPs are subsequently solved using the Finite Element methodology. A mapping connects the solutions of the dual problem to those of the primal problem, facilitating the recovery of the latter. We demonstrate the following ideas:

• For PDEs, the dual equations are, at most, degenerate elliptic BVPs in space-time, allowing for discontinuities in the gradients of the dual fields [3]. This property is exploited in equations such as the Linear Transport and the Burgers equation, where examples illustrate how these gradient discontinuities in the dual field naturally correspond to the discontinuous solutions of the primal IVPs.
• Utilizing the concept of Base States [2, 3], we demonstrate that appropriately selecting them facilitates the solution of the dual problems. The Hamilton-Jacobi form of the Burgers equation utilizes this idea to capture the closest entropy solution among non unique solutions.

Building on the second point, two primary types of algorithms have been developed:

• A Time-Marching Algorithm designed to solve initial value problems by dividing the v total time into smaller intervals/stages. In each stage, a space-time BVP is solved which utilizes the solution obtained in the previous stage.
• Anon-local, nonlinear equation solver with a step-size control (for Newton’s method) is developed which allows obtaining solutions from initial guesses far from a solution.

Wherever applicable, the dual scheme’s performance is evaluated by comparing the primal f ields obtained from dual solutions with the exact primal solutions derived from the given data.