The Poisson equation is central in numerous physics and engineering applications, such as computational fluid dynamics and acoustic wave propagation, where efficient and accurate solutions are essential. This study focuses on the numerical solution of the 2D Poisson equation with Dirichlet boundary conditions using a fourth-order compact Implicit Finite Difference scheme. Finite difference methods, particularly high-order schemes, are advantageous for solving the Poisson equation due to their efficiency and suitability for structured grids. To address the computational demands of large-scale problems, we incorporate domain decomposition and the Multicolor Successive Over Relaxation method, facilitating parallel computation. Through numerical experiments, we demonstrate that our approach significantly enhances both accuracy and computational efficiency when compared to traditional second-order methods.
Abstract The Poisson equation is central in numerous physics and engineering applications, such as computational fluid dynamics and acoustic wave propagation, where efficient and accurate [...]
This work forms the foundation for addressing high-order immersed interface methods to solve interface problems and enables us to conduct in-depth examination of this theory. Here, we focus on the introduction a fourth-order finite-difference formulation to approximate the second-order derivative of discontinuous functions. The approach is based on the combination of a high-order implicit formulation and the immersed interface method. The idea is to modify the standard schemes by introducing additional contribution terms based on jump conditions. These contributions are calculated only at grid points where the stencil intersects with the interface. Here, we discuss the issues of implementing the one-dimensional Poisson equation and the heat conduction equation with discontinuous solutions as a three-point stencil for each grid point on the computational domain. In both cases, the resulting discretization approach yields a tridiagonal linear system with matrix coefficients identical to those employed for smooth solutions. We present several numerical experiments to verify the feasibility and accuracy of the method. Thus, this high-order method provides an attractive numerical framework that can efficiently lead to the solution to more complex problems.
Abstract This work forms the foundation for addressing high-order immersed interface methods to solve interface problems and enables us to conduct in-depth examination of this theory. [...]
This article proposes a parallel implementation using a multicore environment with MPI to the solution of the linear system resulting from the discretization of the Poisson equation in 2D using finite differences and the iterative method of Jacobi. The size of the domain and its corresponding discretization result in a system of linear equations where the number of variables can be millions. The magnitude of the problem allows the algorithm to be highly scalable in parallel; this means that by increasing the number of processors available to solve the system, the execution time will improve considerably. However, as the number of processors increases, the communication work also increases, which stops its performance. Therefore, this article proposes re-engineering the parallel algorithm focused on memory management to speed up its execution and improve its effectiveness.
Abstract This article proposes a parallel implementation using a multicore environment with MPI to the solution of the linear system resulting from the discretization of the Poisson [...]
Multicolor Parallel Fourth-Order Implicit Finite Difference for Solving the 2D Poisson Equation 
