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Lattice Boltzmann methods (LBMs) provide a kinetic simulation technique for solving systems governed by non-linear conservation equations. Most LBMs use the linearised single time relaxation form of the Boltzmann equation to temporally evolve particle distribution functions on a discrete spatial lattice. These kinetic simulation techniques are computationally efficient and highly parallelisable. The use of non-uniform distributions of the spatial grid further enhances the computational efficiency of these algorithms by focusing computational effort around structures of interest such as velocity shocks and current sheets. Here, we apply a non-uniform grid LBM to one-dimensional magnetohydrodynamic systems. Simulations are presented for three sets of initial conditions in order to analyse the effects of the presence of the magnetic induction equation and compare the results to Burger's turbulence. [ABSTRACT FROM AUTHOR]