Numerical analysis of various single-step integrators for parabolic equations

Abstract

Parabolic equations are essential in applications like heat conduction, diffusion processes, and financial modeling. They describe how quantities such as temperature, concentration, or option prices evolve over time, making them crucial in engineering, physics, and economics. This thesis aims to develop efficient numerical methods for solving parabolic problems, particularly in phase-field models, ensuring high accuracy while preserving maximum bound and energy decay properties. In the first part of thesis, we consider the development and analysis of the structure preserving schemes for solving Allen-Cahn equations, that represents an important application of parabolic equations. We apply a k-th order single-step method in time, where the nonlinear term is linearized using multi-step extrapolation. In space, we use a lumped mass finite element method with piecewise r-th order polynomials and Gauss-Lobatto quadrature. At each time level, a cut-off post-processing technique is proposed to eliminate values that violate the maximum bound principle at the finite element nodal points. As a result, the numerical solution satisfies the maximum bound principle at all nodal points, and the optimal error bound O(τk + hr+1) is theoretically proven. These time-stepping schemes include algebraically stable collocation-type methods, which can achieve arbitrarily high order in both space and time. By combining the cut-off strategy with the scalar auxiliary variable (SAV) technique, we develop a class of energy-stable and maximum bound preserving schemes that are arbitrarily high-order in time. In the second part, we present the development and analysis of a class of single-step implicit-explicit schemes for approximately solving linear parabolic equations, which achieves long-time stability and arbitrarily high order in time. This involves splitting the linear operator into symmetric and skew-symmetric components, evaluated implicitly and explicitly, respectively, using the Implicit-Explicit Runge-Kutta Method (IMEX-RK). For the symmetric part, a diagonally implicit method (DIRK) is employed, while the discretization for skew-symmetric part is designed to satisfy the stage orders. This method is applicable to semilinear problems, such as phase-field models, and our analysis is consistent with existing findings, showing energy stability for certain IMEX-RK schemes. Our results reveal intersections up to at least third order, leading to a scheme that preserves both the original energy decay properties and maximum bound principles. In the third part of the thesis, we study the parareal algorithm for solving parabolic equations, which enables parallel-in-time computation and significantly accelerates the process. We prove that the parareal method has a robust convergence rate of about 0.3, provided the ratio J of coarse to fine step size exceeds a certain threshold J*, and the fine propagator meets mild conditions. This convergence is robust even with nonsmooth problem data and boundary condition incompatibilities. Qualified methods include all absolutely stable single-step methods with a stability function satisfying |r(-∞)| < 1, allowing the fine propagator to be of arbitrarily high order. Moreover, we examine popular high-order single-step methods, such as the two-, three-, and four-stage Lobatto IIIC methods, confirming that their corresponding parareal algorithms converge linearly with a factor of 0.31, with a threshold J* = 2. At the end of each chapter, we present numerical results that support the theoretical findings and inspire future investigations.