Higher-order generalized-$α$ methods for parabolic problems

Abstract

We propose a new class of high-order time-marching schemes with dissipation user-control and unconditional stability for parabolic equations. High-order time integrators can deliver the optimal performance of highly-accurate and robust spatial discretizations such as isogeometric analysis. The generalized-α method delivers unconditional stability and second-order accuracy in time and controls the numerical dissipation in the discrete spectrum's high-frequency region. Our goal is to extend the generalized-alpha methodology to obtain a high-order time marching methods with high accuracy and dissipation in the discrete high-frequency range. Furthermore, we maintain the stability region of the original, second-order generalized-alpha method foe the new higher-order methods. That is, we increase the accuracy of the generalized-α method while keeping the unconditional stability and user-control features on the high-frequency numerical dissipation. The methodology solve k > 1, k ∈ N matrix problems and updates the system unknowns, which correspond to higher-order terms in Taylor expansions to obtain (3/2k) th -order method for even k and (3/2k + 1/2) th -order for odd k. A single parameter ρ ∞ controls the dissipation, and the update procedure follows the formulation of the original secondorder method. Additionally, we show that our method is A-stable and setting ρ ∞ = 0 allows us to obtain an L-stable method. Lastly, we extend this strategy to analyze the accuracy order of a generic method.

A note on the conservation properties of the generalized-$α$ method

“…on a uniform temporal mesh where U + +1∕2 ∶= U +1 + − 1 2 Δ Ü +1 and U + −1∕2 = U + − 1 2 Δ Ü , and as a result, application of the second-order generalized-method to a Galerkin elastodynamics semi-discretization results in a fully-discrete method with a discrete balance law for momentum. We leave further analysis of this, as well as extension of the theoretical results shown here to higher-order generalizations of the generalized-method 27,28 , for future work.…”

The effect of time discretization on the solution of parabolic PDEs with ANNs

“…where L is intended as a linear elliptic operator acting only on the spatial variables, B is a boundary operator and f, g are given source and boundary data. We are interested in the numerical resolution of equation (1), in particular in a solution that can be written as an ANN of ELM-type in space:…”

“…The last equation can be regarded as an elliptic equation. 1 The righthand side consists of given functions and quantities that can be explicitly evaluated. The first class is more natural for problems where the diffusion is dominated by the transport, leading to hyperbolic-type behaviors, or where stiffness arises in time.…”

“…Moreover, the first class of methods can be profitably applied when the matrix involved in the problem has a structure that can be exploited. The introduction and study of different time-marching procedures has been proposed also for the parabolic case, for both traditional methods [16,17] and newer ones, such as Isogeometric Analysis [1,2,14] and Physics-Informed Neural Networks [9].…”