Before revision:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which has been studied previously only in the diffusive setting , and prove that it is second order accurate in the size of the time discretization, making it the first of its kind.
After revision:  Further, we introduce a new algorithm in this setting, the weak trapezoidal algorithm, which has been studied previously only in the diffusive setting , and prove that the leading order of the error process scales with the square of the time discretization, making it the first of its kind.