In this work we present a framework for the construction of robust a posteriori
estimates for classes of finite difference schemes. We are motivated by the relative
lack of such frameworks compared to existing ones for other numerical discretisation
methods, such as finite elements and finite volumes.
The framework we propose is based on the use of reconstructions, which are
obtained by post-processing the finite difference solution. The post-processed object
is a key ingredient in obtaining a posteriori bounds using the relevant stability
framework of the problem. The resulting bounds are fully computable and allow us
to establish a posteriori error control over the problem at hand.
In the first part of the thesis we motivate and investigate the behaviour of our
framework using model ODE, elliptic and hyperbolic problems. We use our framework to obtain reconstructions which are used to compute a posteriori error estimates. We validate the numerical behaviour of these estimates using solutions of
varying regularity.
In the second part of the thesis we focus on hyperbolic conservation laws in one
spatial dimension and we deal with scalar problems as well as systems. Hyperbolic
conservation laws are widely used in the modelling of physical phenomena. The
numerical modelling of conservation laws, which arises due to the frequent lack
of explicit solutions, is challenging, largely due to the complex behaviour these
problems exhibit, such as shock formation even with smooth initial conditions.
In this setting, we present a framework which is applicable to general non-linear
conservation laws. We investigate its numerical behaviour and showcase our results
by using popular finite difference discretisations for a range of problems.
We demonstrate that the the framework can produce optimal estimates, capable
of tracking features of interest and act as refinement/coarsening indicators
