Abstract: We describe numerical methods for computing solutions of the unsteady transonic small disturbance equations that describe the Mach reflection of weak shock waves. We solve the equations in self-similar variables and use local grid refinement to resolve the solution in the reflection region. The solutions contain a complex structure consisting of a sequence of triple points and tiny supersonic patches directly behind the leading triple point, formed by the reflection of weak shock and expansion waves between the sonic line and the Mach shock. The presence of an expansion fan at each triple point resolves the von Neumann triple point paradox. Additionally, we will present some self-similar solutions for the reflection of expansion waves.