Preserving energy resp. dissipation in numerical PDEs using the Average Vector Field method

https://www.ece.uvic.ca/~bctill/papers/numacoust/Hong_Li_2006.pdf

Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations

This chapter surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical methods and their analyses, the chapter presents a holistic overview about the main ideas of phase field modelling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state of the art of numerical approximations of various phase field models with an emphasis on discussing the main ideas of numerical analysis techniques. The chapter also reviews recent development on adaptive grid methods and various applications of the phase field modelling and their numerical methods in materials science, fluid mechanics, biology and image science.

The phase field method for geometric moving interfaces and their numerical approximations

http://hans.munthe-kaas.no/work/Blog/Entries/2006/1/2_Article:_Globally_conservative_properties_and_error_estimation_of_a_multi-symplectic_scheme_.._files/hong06gcp.pdf

Arieh Iserles

Papers

A First Course in the Numerical Analysis of Differential Equations: Gaussian elimination for sparse linear equations