Journal of Computational Physics
SIAM Journal on Scientific and Statistical Computing
Implicit-explicit methods for time-dependent partial differential equations
SIAM Journal on Numerical Analysis
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
Applied Numerical Mathematics - Special issue on time integration
On the stability of implicit-explicit linear multistep methods
Applied Numerical Mathematics - Special issue on time integration
Analysis of stiffness in the immersed boundary method and implications for time-stepping schemes
Journal of Computational Physics
Linearly implicit Runge—Kutta methods for advection—reaction—diffusion equations
Applied Numerical Mathematics
The Method of Regularized Stokeslets
SIAM Journal on Scientific Computing
Additive Runge-Kutta schemes for convection-diffusion-reaction equations
Applied Numerical Mathematics
Multi-adaptive time integration
Applied Numerical Mathematics - Special issue: Workshop on innovative time integrators for PDEs
Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics
Journal of Computational Physics
Accelerating the convergence of spectral deferred correction methods
Journal of Computational Physics
Unconditionally stable discretizations of the immersed boundary equations
Journal of Computational Physics
A stochastic immersed boundary method for fluid-structure dynamics at microscopic length scales
Journal of Computational Physics
2-D Parachute Simulation by the Immersed Boundary Method
SIAM Journal on Scientific Computing
Error Analysis of IMEX Runge-Kutta Methods Derived from Differential-Algebraic Systems
SIAM Journal on Numerical Analysis
Multirate Runge-Kutta schemes for advection equations
Journal of Computational and Applied Mathematics
Modeling slender bodies with the method of regularized Stokeslets
Journal of Computational Physics
A Hybrid Implicit-Explicit Adaptive Multirate Numerical Scheme for Time-Dependent Equations
Journal of Scientific Computing
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The method of regularized Stokeslets is a numerical approach to approximating solutions of fluid-structure interaction problems in the Stokes regime. Regularized Stokeslets are fundamental solutions to the Stokes equations with a regularized point-force term that are used to represent forces generated by a rigid or elastic object interacting with the fluid. Due to the linearity of the Stokes equations, the velocity at any point in the fluid can be computed by summing the contributions of regularized Stokeslets, and the time evolution of positions can be computed using standard methods for ordinary differential equations. Rigid or elastic objects in the flow are usually treated as immersed boundaries represented by a collection of regularized Stokeslets coupled together by virtual springs which determine the forces exerted by the boundary in the fluid. For problems with boundaries modeled by springs with large spring constants, the resulting ordinary differential equations become stiff, and hence the time step for explicit time integration methods is severely constrained. Unfortunately, the use of standard implicit time integration methods for the method of regularized Stokeslets requires the solution of dense nonlinear systems of equations for many relevant problems. Here, an alternate strategy using an explicit multirate time integration scheme based on spectral deferred corrections is incorporated that in many cases can significantly decrease the computational cost of the method. The multirate methods are higher-order methods that treat different portions of the ODE explicitly with different time steps depending on the stiffness of each component. Numerical examples on two nontrivial three-dimensional problems demonstrate the increased efficiency of the multi-explicit approach with no significant increase in numerical error.