In this work, we present the application of a newly developed implicit second-order Material Point Method (MPM) on offshore geotechnical applications. The presented second-order MPM uses a special set of piecewise quadratic shape functions to circumvent the well-known issue of producing zero nodal mass contributions. To mitigate the effect of the standard MPM to produce highly oscillating stresses across cell interfaces, we have carried over our ideas obtained from the derivation of the second-order MPM to the Dual Domain Material Point (DDMP) Method, too. The resulting second-order DDMP Method produces a smoother stress distribution across the entire computational domain while being able to profit from the improved convergence rates of second-order finite elements. In a numerical example from geotechnical engineering applications, we illustrate the practical application of our enhanced Material Point and DDMP Methods by simulating a cone penetration. @en