2019/11/19 by Marcin Łoś, Pouria Behnoudfar, Los, Marcin +5
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1911.08158
openalex publication_date 2019/11/19 · openalex created_date 2019/12/05 · openalex updated_date 2026/07/28
We use the alternating direction method to simulate implicit dynamics. ur spatial discretization uses isogeometric analysis. Namely, we simulate a (hyperbolic) wave propagation problem in which we use tensor-product B-splines in space and an implicit time marching method to fully discretize the problem. We approximate our discrete operator as a Kronecker product of one-dimensional mass and stiffness matrices. As a result of this algebraic transformation, we can factorize the resulting system of equations in linear (i.e., O(N)) time at each step of the implicit method. We demonstrate the performance of our method in the model P-wave propagation problem. We then extend it to simulate the linear elasticity problem once we decouple the vector problem using alternating triangular methods. We proof theoretically and experimentally the unconditional stability of both methods.