  We propose a variational scheme for computing Wasserstein gradient flows. The
scheme builds upon the Jordan--Kinderlehrer--Otto framework with the
Benamou-Brenier's dynamic formulation of the quadratic Wasserstein metric and
adds a regularization by the Fisher information. This regularization can be
derived in terms of energy splitting and is closely related to the
Schr{\"o}dinger bridge problem. It improves the convexity of the variational
problem and automatically preserves the non-negativity of the solution. As a
result, it allows us to apply sequential quadratic programming to solve the
sub-optimization problem. We further save the computational cost by showing
that no additional time interpolation is needed in the underlying dynamic
formulation of the Wasserstein-2 metric, and therefore, the dimension of the
problem is vastly reduced. Several numerical examples, including porous media
equation, nonlinear Fokker-Planck equation, aggregation diffusion equation, and
Derrida-Lebowitz-Speer-Spohn equation, are provided. These examples demonstrate
the simplicity and stableness of the proposed scheme.
