Arbitrarily High-Order Unconditionally Energy Stable Schemes for Thermodynamically Consistent Gradient Flow Models

We present a systematic approach to developing arbitrarily high-order, unconditionally energy stable numerical schemes for thermodynamically consistent gradient flow models that satisfy energy dissipation laws. Utilizing the energy quadratization method, we formulate the gradient flow model into an equivalent form with a corresponding quadratic free energy functional. Based on the equivalent form with a quadratic energy, we propose two classes of energy stable numerical approximations. In the first approach, we use a prediction-correction strategy to improve the accuracy of linear numerical schemes. In the second approach, we adopt the Gaussian collocation method to discretize the equivalent form with a quadratic energy, arriving at an arbitrarily high-order scheme for gradient flow models. Schemes derived using both approaches are proved rigorously to be unconditionally energy stable. The proposed schemes are then implemented in four gradient flow models numerically to demonstrate their accuracy and effectiveness. Detailed numerical comparisons among these schemes are carried out as well. These numerical strategies are rather general so that they can be readily generalized to solve any thermodynamically consistent PDE models

Fully Discrete Second-Order Linear Schemes for Hydrodynamic Phase Field Models of Binary Viscous Fluid Flows with Variable Densities

We develop spatial-temporally second-order, energy stable numerical schemes for two classes of hydrodynamic phase field models of binary viscous fluid mixtures of different densities. One is quasi-incompressible while the other is incompressible. We introduce a novel energy quadratization technique to arrive at fully discrete linear schemes, where in each time step only a linear system needs to be solved. These schemes are then shown to be unconditionally energy stable rigorously subject to periodic boundary conditions so that a large time step is plausible. Both spatial and temporal mesh refinements are conducted to illustrate the second-order accuracy of the schemes. The linearization technique developed in this paper is so general that it can be applied to any thermodynamically consistent hydrodynamic theories so long as their energies are bounded below. Numerical examples on coarsening dynamics of two immiscible fluids and a heavy fluid drop settling in a lighter fluid matrix are presented to show the effectiveness of the proposed linear schemes. Predictions by the two fluid mixture models are compared and discussed, leading to our conclusion that the quasiincompressible model is more reliable than the incompressible one

Fully Discrete Second-Order Linear Schemes for Hydrodynamic Phase Field Models of Binary Viscous Fluid Flows with Variable Densities

We develop spatial-temporally second-order, energy stable numerical schemes for two classes of hydrodynamic phase field models of binary viscous fluid mixtures of different densities. One is quasi-incompressible while the other is incompressible. We introduce a novel energy quadratization technique to arrive at fully discrete linear schemes, where in each time step only a linear system needs to be solved. These schemes are then shown to be unconditionally energy stable rigorously subject to periodic boundary conditions so that a large time step is plausible. Both spatial and temporal mesh refinements are conducted to illustrate the second-order accuracy of the schemes. The linearization technique developed in this paper is so general that it can be applied to any thermodynamically consistent hydrodynamic theories so long as their energies are bounded below. Numerical examples on coarsening dynamics of two immiscible fluids and a heavy fluid drop settling in a lighter fluid matrix are presented to show the effectiveness of the proposed linear schemes. Predictions by the two fluid mixture models are compared and discussed, leading to our conclusion that the quasiincompressible model is more reliable than the incompressible one