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Stiff systems of ordinary differential equations (ODEs) are pervasive in many science and engineering fields, yet standard neural ODE approaches struggle to learn them. This limitation is the main barrier to the widespread adoption of…
In this paper, we extend the Paired-Explicit Runge-Kutta schemes by Vermeire et. al. to fourth-order of consistency. Based on the order conditions for partitioned Runge-Kutta methods we motivate a specific form of the Butcher arrays which…
We present a novel and general methodology for building second-order finite volume implicit-explicit Runge-Kutta numerical schemes for solving two-dimensional financial parabolic PDEs with mixed derivatives. The methods achieve second-order…
The task of integrating a large number of independent ODE systems arises in various scientific and engineering areas. For nonstiff systems, common explicit integration algorithms can be used on GPUs, where individual GPU threads…
Discontinuous Galerkin (DG) methods for hyperbolic partial differential equations (PDEs) with explicit time-stepping schemes, such as strong stability-preserving Runge-Kutta (SSP-RK), suffer from time-step restrictions that are…
We construct a family of embedded pairs for optimal strong stability preserving explicit Runge-Kutta methods of order $2 \leq p \leq 4$ to be used to obtain numerical solution of spatially discretized hyperbolic PDEs. In this construction,…
The recently developed technique of DOC kernels has been a great success in the stability and convergence analysis for BDF2 scheme with variable time steps. However, such an analysis technique seems not directly applicable to problems with…
The use of high order fully implicit Runge-Kutta methods is of significant importance in the context of the numerical solution of transient partial differential equations, in particular when solving large scale problems due to fine space…
We propose a better method to determine the stability region of an L-stable implicit-explicit Runge-Kutta scheme. This method always provides the correct result, while other methods sometimes give wrong result. It is useful in the analysis…
Efficient long-time integration of nonlinear fractional differential equations is significantly challenging due to the integro-differential nature of the fractional operators. In addition, the inherent non-smoothness introduced by the…
Explicit stabilized methods are highly efficient time integrators for large and stiff systems of ordinary differential equations especially when applied to semi-discrete parabolic problems. However, when local spatial mesh refinement is…
In this paper, we develop a class of high-order conservative methods for simulating non-equilibrium radiation diffusion problems. Numerically, this system poses significant challenges due to strong nonlinearity within the stiff source terms…
Strong stability preserving (SSP) Runge-Kutta methods are desirable when evolving in time problems that have discontinuities or sharp gradients and require nonlinear non-inner-product stability properties to be satisfied. Unlike the case…
Systems driven by multiple physical processes are central to many areas of science and engineering. Time discretization of multiphysics systems is challenging, since different processes have different levels of stiffness and characteristic…
As a popular and easy-to-implement machine learning method for solving differential equations, the physics-informed neural network (PINN) sometimes may fail and find poor solutions which bias against the exact ones. In this paper, we…
High order strong stability preserving (SSP) time discretizations ensure the nonlinear non-inner-product strong stability properties of spatial discretizations suited for the stable simulation of hyperbolic PDEs. Over the past decade…
This paper investigates, a new class of fractional order Runge-Kutta (FORK) methods for numerical approximation to the solution of fractional differential equations (FDEs). By using the Caputo generalizedTaylor formula and the total…
We present a dynamically load-balanced parallel $ p $-adaptive implicit high-order flux reconstruction method for under-resolved turbulence simulation. The high-order explicit first stage, singly diagonal implicit Runge-Kutta (ESDIRK)…
Segregated Runge-Kutta (SRK) schemes are time integration methods for the incompressible Navier-Stokes equations. In this approach, convection and diffusion can be independently treated either explicitly or implicitly, which in particular…
The pressure-correction method is a well established approach for simulating unsteady, incompressible fluids. It is well-known that implicit discretization of the time derivative in the momentum equation e.g. using a backward…