Date of Award

2026-05-01

Degree Name

Master of Science

Department

Computational Science

Advisor(s)

Peimeng Yin

Abstract

Time-dependent fourth-order partial differential equations arise in a wide range of applications in applied mathematics, physics, and engineering, including thin structure models, phase separation, and pattern formation. Their numerical approximation is challenging because the presence of fourth-order spatial derivatives typically requires high-regularity discretizations. A useful alternative is to reformulate the original problem as a coupled second-order system and approximate it by mixed discontinuous Galerkin (DG) methods.

This thesis builds upon the penalty-free mixed DG framework developed by Liu and Yin for time-dependent fourth-order problems. That framework avoids the use of interior penalty parameters, preserves the symmetry of the associated bilinear form, and yields semi-discrete \(L^2\) stability together with optimal spatial convergence. However, in its original form, the temporal discretization is based on the Crank--Nicolson method, whose accuracy is limited to second order in time.

The main goal of this thesis is to extend this penalty-free mixed DG framework by replacing the Crank--Nicolson discretization with algebraically stable implicit Runge Kutta methods. This leads to a fully discrete Runge Kutta discontinuous Galerkin (RKDG) scheme for time-dependent fourth-order linear operators. The resulting formulation combines the penalty-free mixed DG structure in space with high-order implicit time integration.

A general Runge Kutta time discretization framework is developed and then specialized toimplicit schemes, with particular emphasis on diagonally implicit methods of SDIRK type for implementation purposes. Under algebraic stability assumptions on the Runge Kutta coefficients, the fully discrete mixed RKDG method is shown to inherit the dissipative structure of the semi-discrete formulation and to satisfy unconditional \(L^2\) stability. In addition, an optimal fully discrete error estimate of the form \[ \|u(\cdot,t^n)-u_h^n\| \le C\bigl(h^{k+1} + \Delta t^p\bigr) \] is derived, where \(k\) denotes the polynomial degree of the DG approximation and \(p\) is the order of the Runge Kutta method.

The thesis also discusses extensions of the RKDG framework to more general fourth-order linear operators, multidimensional settings, and possible nonlinear energy-dissipative models. Altogether, the results establish a systematic penalty-free RKDG framework for the stable and high-order time integration of fourth-order evolution equations.

Language

en

Provenance

Received from ProQuest

File Size

75 p.

File Format

application/pdf

Rights Holder

Jose Armando Perez Becerra

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