The Poisson-Boltzmann (PB) equation is an established multiscale model for electrostatic analysis of biomolecules and other dielectric systems. PB based molecular dynamics (MD) approach has a potential to tackle large biological systems. Obstacles that hinder the current development of PB based MD methods are concerns in accuracy, stability, efficiency and reliability. The presence of complex solvent-solute interface, geometric singularities and charge singularities leads to challenges in the numerical solution of the PB equation and electrostatic force evaluation in PB based MD methods. Recently, the matched interface and boundary (MIB) method has been utilized to develop the first second order accurate PB solver that is numerically stable in dealing with discontinuous dielectric coefficients, complex geometric singularities and singular source charges. The present work develops the PB based MD approach using the MIB method. New formulation of electrostatic forces is derived to allow the use of sharp molecular surfaces. Accurate reaction field forces are obtained by directly differentiating the electrostatic potential. Dielectric boundary forces are evaluated at the solvent-solute interface using an accurate Cartesian-grid surface integration method. The electrostatic forces located at reentrant surfaces are appropriately assigned to related atoms. Extensive numerical tests are carried out to validate the accuracy and stability of the present electrostatic force calculation. The new PB based MD method is implemented in conjunction with the AMBER package. MIB based MD simulations of biomolecules are demonstrated via a few example systems.