Abstract : Hyperbolic partial differential equations and their numerical solutions play an important role in several fields of applied mathematics. Many interesting applications of related PDEs are stiff in nature, so implicit time discretizations with enhanced stability properties are good candidates for their numerical solution. The minisymposium shall discuss important aspects of such methods like higher order accuracy, non-oscillatory behavior, well-balancing, asymptotic-preserving, efficient solvers, and combinations with explicit schemes.
Organizer(s) : Peter Frolkovič, Pep Mulet, Carlos Parés
[04131] Semi-implicit schemes for a convection-diffusion-reaction model of sequencing batch reactors
Format : Online Talk on Zoom
Author(s) :
Raimund Bürger (Universidad de Concepción)
Julio Careaga (Radboud University)
Stefan Diehl (Lund University)
Romel Pineda (Universidad de Concepción)
Abstract : Sequencing batch reactors are used in wastewater treatment for the settling of solid biomass particles simultaneously with biochemical reactions with nutrients dissolved in the liquid. This unit can be modeled as a moving-boundary problem for a degenerating convection-diffusion-reaction system. This model is transformed to a fixed computational domain and is discretized by an explicit and semi-implicit monotone schemes. Both variants obey an invariant region property. Numerical examples illustrate that the semi-implicit variant is more efficient.
[01691] High resolution well-balanced compact implicit numerical scheme for numerical solution of the shallow water equations
Format : Talk at Waseda University
Author(s) :
Michal Žeravý (Slovak University of Technology in Bratislava)
Peter Frolkovič (Slovak University of Technology in Bratislava)
Abstract : In this talk, we deal with the numerical solution of shallow water equations with topography in one-dimensional case using a high resolution well-balanced compact implicit numerical scheme. The upwind scheme uses the fractional step method with the fast sweeping method. Consequently, the Jacobian matrix of the discrete system of nonlinear algebraic equations is always either a lower or upper triangular matrix.
[01692] Numerical solution of scalar hyperbolic problems using the third order accurate compact implicit scheme
Format : Talk at Waseda University
Author(s) :
Dagmar Zakova (Slovak University of Technology in Bratislava)
Peter Frolkovic (Slovak University of Technology in Bratislava)
Abstract : This work presents compact implicit numerical schemes for solving scalar hyperbolic problems. We propose details of the third-order accurate scheme using the finite volume method. To avoid unphysical oscillations in the case of nonsmooth solution, we modify the scheme using ENO and WENO approximation in space. Applications to one-dimensional conservation laws are shown.
[05425] High-fidelity multiderivative time integration for compressible flows
Format : Talk at Waseda University
Author(s) :
Arjun Thenery Manikantan (Hasselt University)
Jochen Schütz (Hasselt University)
Jonas Zeifang (Hasselt University)
Abstract : We present a high-order parallel-in-time implicit multiderivative time-stepping scheme for Navier Stokes equations spatially discretized with the Discontinuous Galerkin Spectral element method. The time-stepping starts with a predicted solution, and the correction steps increase the order of accuracy up to the order of the underlying quadrature rule. Implicit non-linear(linear) equations are solved using Newton's and the general minimal residual methods. Extensive numerical results are shown for convergence, parallel efficiency, and other scheme features.
[01684] Implicit and semi-implicit well-balanced finite volume methods for general 1d systems of balance laws
Format : Talk at Waseda University
Author(s) :
Carlos Parés (University of Málaga)
Irene Gómez-Bueno (University of Málaga)
Manuel Jesús Castro (University of Málaga)
Sebastiano Boscarino (University of Catania)
Giovanni Russo (University of Catania)
Abstract : In this work a general family of implicit and semi-implicit well-balanced finite volume numerical methods for nonlinear hyperbolic systems of balance laws will be presented. These methods are obtained by extending a general strategy introduced by some of the authors to design high-order well-balanced explicit methods. This strategy, based on the computation of local steady states, will be combined with implicit RK or IMEX solvers in time. Different applications will be shown.
[01665] SHALLOW-WATER MODEL: IMPLICIT FULLY WELL-BALANCED METHODS IN THE LAGRANGE-PROJECTION FRAMEWORK
Format : Talk at Waseda University
Author(s) :
Celia Caballero Cárdenas (Universidad de Málaga)
Manuel Jesús Castro Díaz (Universidad de Málaga)
Tomás Morales de Luna (Universidad de Malaga)
María de la Luz Muñoz-Ruiz (Universidad de Málaga)
Abstract : We propose fully well-balanced Lagrange-Projection finite volume schemes for the shallow-water model. This two-step approach separates acoustic and transport phenomena, allowing for implicit-explicit and large time step schemes with CFL restriction based on slower transport waves.
[01664] Hyperbolic systems with stiff relaxation: asymptotic-preserving and well-balanced schemes
Format : Talk at Waseda University
Author(s) :
Irene Gómez-Bueno (University of Málaga)
Sebastiano Boscarino (University of Catania)
Manuel Jesús Castro Díaz (University of Málaga)
Carlos Parés (University of Málaga)
Giovanni Russo (University of Catania)
Abstract : We consider hyperbolic systems depending on a stiff parameter $\varepsilon$: when $\varepsilon$ is small, numerical schemes may produce spurious results. Implicit-explicit Runge–Kutta schemes have been widely used for their time evolution. Our goal is to design high-order asymptotic-preserving methods which are at the same time well-balanced for the asymptotic limit system.
[01673] MIRK methods and applications in RRMHD and neutrino transport equations
Format : Talk at Waseda University
Author(s) :
Isabel Cordero-Carrión (University of Valencia)
Samuel Santos-Pérez (University of Valencia)
Martin Obergaulinger (University of Valencia)
Abstract : We present the Minimally-Implicit Runge-Kutta $($MIRK$)$ methods for the numerical resolution of hyperbolic equations with stiff source terms. We apply these schemes to the resistive relativistic magnetohydrodynamic $($RRMHD$)$ and the M1 neutrino transport equations. Previous approaches rely on Implicit-Explicit Runge-Kutta schemes. The MIRK methods are able to deal with stiff terms producing stable numerical evolutions and their computational cost is similar to the standard explicit methods.
[03904] Implicit-explicit schemes for Cahn-Hilliard-Navier-Stokes equations
Format : Talk at Waseda University
Author(s) :
Pep Mulet (University of Valencia)
Abstract : Navier-Stokes-Cahn-Hilliard equations are a system of fourth-order partial differential equations that model the evolution of compressible mixtures of binary fluids (e.g. foams, solidification processes, fluid–gas interface) under gravitational effects. Our aim is to use implicit-explicit time-stepping schemes to avoid the severe restriction posed by the high order terms for the efficient numerical solution of problems with these equations.
