Pedro Henrique de Almeida Konzen

Pedro Henrique de Almeida Konzen, Esequia Sauter, Fabio Souto de Azevedo et al.

In this paper we present a simple and accurate second order finite element scheme to simulate the Burgers' equation on the whole real line and subjected to initial conditions with compact support. The numerical simulations are performed by considering a sequence of auxiliary spatially dimensionless Dirichlet's problems parameterized by the domain's semidiameter L. Gaining advantage from the well-known convective-diffusive effects of the Burgers' equation, computations start by choosing L larger than the semidiameter of the support of the initial condition and, as solution diffuses out, L is increased appropriately. By direct comparisons between numerical and analytic solutions and its asymptotic behavior, we conclude this simple scheme is very accurate and can be applied to numerically investigate properties of this and similar equations on infinite domains.

Vitória Biesek, Pedro Henrique de Almeida Konzen

The Burgers equation is a well-established test case in the computational modeling of several phenomena such as fluid dynamics, gas dynamics, shock theory, cosmology, and others. In this work, we present the application of Physics-Informed Neural Networks (PINNs) with an implicit Euler transfer learning approach to solve the Burgers equation. The proposed approach consists in seeking a time-discrete solution by a sequence of Artificial Neural Networks (ANNs). At each time step, the previous ANN transfers its knowledge to the next network model, which learns the current time solution by minimizing a loss function based on the implicit Euler approximation of the Burgers equation. The approach is tested for two benchmark problems: the first with an exact solution and the other with an alternative analytical solution. In comparison to the usual PINN models, the proposed approach has the advantage of requiring smaller neural network architectures with similar accurate results and potentially decreasing computational costs.