AbstractArbitrary Lagrangian–Eulerian (ALE) methods are commonly used for challenging problems in hydrodynamics. Among the most challenging matters are the approximations in the presence of multiple materials. The ALE code ALEGRA has used a constant volume method for computing the impact of multiple materials on both the Lagrangian step and the remap step of the method. Here, we describe modifications to these methods that provide greater modeling fidelity and better numerical and computational performance. In the Lagrangian step, the effects of differences in material response were not included in the constant volume method, but have been included in the new method. The new methodology can produce unstable results unless the changes in the variable states are carefully controlled. Both the stability analysis and the control of the instability are described. In the standard (Van Leer) method for the remap, the numerical approximation did not account for the presence of a material interface directly. The new methodology uses different, more stable and more dissipative numerical approximations in and near material interfaces. In addition, the standard numerical method, which is second‐order accurate, has been replaced by a more accurate method, which is third‐order accurate in one dimension. Published in 2010 by John Wiley & Sons, Ltd.