# Registered Data

## [00779] Advances in numerical methods for evolutionary PDEs and applications

**Session Date & Time**:- 00779 (1/3) : 5B (Aug.25, 10:40-12:20)
- 00779 (2/3) : 5C (Aug.25, 13:20-15:00)
- 00779 (3/3) : 5D (Aug.25, 15:30-17:10)

**Type**: Proposal of Minisymposium**Abstract**: Several models in science, physics and engineering, are described by evolutionary systems of partial differential equations (PDEs). The purpose of the MS is to gather researcher interested in the development of innovative techniques for the numerical solution of a wide class of evolutionary problems, in several contexts: kinetic theory of rarefied gases, linear and nonlinear waves, viscoelasticity, multiphase flows, radiation hydrodynamics, traffic flows, shallow water, just to mention some examples. The mini-symposium will deal with several issues related to the numerical solution of such equations, including, among others, multi-scale issues, asymptotic preserving schemes, high order discretization in space and time, and stability analysis.**Organizer(s)**: Sebastiano Boscarino, Giuseppe Izzo, Giovanni Russo**Classification**:__65M06__,__65M08__,__65M12__,__65M20__,__65M22__**Speakers Info**:- Ben Scott Southworth (Los Alamos National Laboratory)
- Jie Shen ( Purdue University)
- Simone Chiocchetti (University of Trento)
- Tao Xiong (Xiamen University)
- Peter Frolkovic (Slovak University od Technology)
- Umberto Zerbinati (University of Oxford)
- Clarissa Astuto (King Abdullah University of Science and Technology (KAUST))
- Guoliang Zhang (Shanghai Jiaotong University)
- Seung Yeon Cho (Gyeongsang National University)
- David Ketcheson (King Abdullah University of Science and Technology (KAUST))
- Lukas Einkemmer (University of Innsbruck)
**Sebastiano Boscarino**(University of Catania, Italy)

**Talks in Minisymposium**:**[03164] Asymptotic preserving and uniformly unconditionally stable schemes for kinetic transport equations****Author(s)**:**Guoliang Zhang**( Shanghai Jiaotong University)

**Abstract**: In this talk, we will give uniformly unconditionally stable finite difference schemes for kinetic transport equations in the diffusive scaling. The schemes are based on a coupling of macroscopic and microscopic equations, by utilizing a backward semi-Lagrangian approach for transport part, and implicit method for the diffusive part. The schemes can be shown to be asymptotic preserving in the diffusive limit. Uniformly unconditional stabilities are verified by Fourier analysis. Numerical experiments will demonstrate their good performances.

**[03444] Semi-implicit numerical methods for level set equations****Author(s)**:- Nikola Gajdošová (Slovak University of Technology in Bratislava)
- Katarína Lacková (Slovak University of Technology in Bratislava)
**Peter Frolkovič**(Slovak University of Technology in Bratislava)

**Abstract**: We present semi-implicit higher order numerical methods to solve nonlinear level set equations for evolving interfaces. We introduce up to third order accurate unconditionally stable numerical schemes to solve advection by external velocity and speed in normal direction, and, eventually, regularized by a small curvature term. The methods have fully upwinded stencil in its implicit part so efficient algebraic solvers like fast sweeping methods can be applied.

**[03846] Efficient simulation of high-dimensional kinetic equation****Author(s)**:**Lukas Einkemmer**(University of Innsbruck)

**Abstract**: Solving high-dimensional kinetic equations (such as the Vlasov equation or the Boltzmann equations) numerically is extremely challenging. Methods that discretize phase space suffer from the exponential growth of the number of degrees of freedom, the so-called curse of dimensionality, while Monte Carlo methods converge slowly and suffer from numerical noise. In addition, standard complexity reduction techniques (such as sparse grids) usually perform rather poorly due to the lack of smoothness for such problems. Dynamical low-rank techniques approximate the dynamics by a set of lower-dimensional objects. For those low-rank factors, partial differential equations are derived that can then be solved numerically. We will show that such dynamical low-rank approximations work well for a range of kinetic equations due to their capacity to handle non-smooth solutions and the fact that in many situations important physical limit regimes are represented very efficiently by such an approximation (e.g. fluid or diffusive limits).

**[03910] High-order semi-implicit schemes for evolutionary partial differential equations with higher order derivatives****Author(s)**:**Sebastiano Boscarino**(University of Catania, Italy)

**Abstract**: The aim of this work is to apply a semi-implicit (SI) strategy in an implicit-explicit (IMEX) Runge-Kutta (RK) setting introduced in (S. Boscarino- F. Filbet, G. Russo, JSC 2016) to a sequence of 1D time-dependent partial differential equations (PDEs) with high order spatial derivatives. This strategy gives a great flexibility to treat these equations, and allows the construction of simple linearly implicit schemes without any Newton’s iteration. Furthermore, the SI IMEX- RK schemes so designed does not need any severe time step restriction that usually one has using explicit methods for the stability, i.e. ∆t = O(∆t^k) for the k-th (k ≥ 2) order PDEs. For the space discretization, this strategy is combined with finite differ- ence schemes. We illustrate the effectiveness of the schemes with many applications to dissipative, dispersive and biharmonic-type equations. Numerical experiments show that the proposed schemes are stable and can achieve optimal orders of accuracy.

**[04458] New highly stiff-stable schemes for linear and nonlinear parabolic equations****Author(s)**:**JIE SHEN**(Purdue University)

**Abstract**: We construct a class of new highly stiff-stable schemes for linear and nonlinear parabolic equations based on Taylor expansions at time $t_{n+k}$ where $k\ge 1$ is a tunable parameter. We show that their numerical solutions are bounded unconditionally (resp. for sufficiently small time steps) for linear (resp. nonlinear ) parabolic equations, and derive their optimal error estimates for a large class of nonlinear parabolic equations. We also present numerical results to show the advantages of the new schemes compared with the classic IMEX schemes based on Taylor expansions at time $t_{n+1}$.

**[04700] High order structure preserving schemes for MHD flows in all sonic Mach numbers****Author(s)**:**Tao Xiong**(Xiamen University)

**Abstract**: In this work, a high-order semi-implicit (SI) asymptotic preserving (AP) and divergence-free finite difference weighted essentially nonoscillatory (WENO) scheme is proposed for magnetohydrodynamic (MHD) equations. We consider the sonic Mach number $\varepsilon$ ranging from $0$ to $\mathcal{O}(1)$. High-order accuracy in time is obtained by SI implicit-explicit Runge–Kutta (IMEX-RK) time discretization. High-order accuracy in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions. A constrained transport method is applied to maintain a discrete divergence-free condition. We formally prove that the scheme is AP. Asymptotic accuracy (AA) in the incompressible MHD limit is obtained if the implicit part of the SI IMEX-RK scheme is stiffly accurate. Numerical experiments are provided to validate the AP, AA, and divergence-free properties of our proposed approach. Besides, the scheme can well capture discontinuities such as shocks in an essentially non-oscillatory fashion in the compressible regime, while it is also a good incompressible solver with uniform large-time step conditions in the low sonic Mach limit.

**[04882] Asymptotic preserving scheme for ExB drift****Author(s)**:**Umberto Zerbinati**(University of Oxford)- Giovanni Russo (University of Catania)

**Abstract**: In this talk, we explore an asymptotic preserving scheme for ExB drift. The key idea behind the scheme here presented is to treat the highly oscillatory component of the velocity using an exponential integrator. We will apply this numerical scheme to particles in cell plasma simulation under the effect of a strong constant magnetic field.

**[04978] A conservative semi-Lagrangian method for inhomogeneous Boltzmann equation****Author(s)**:**Seung Yeon Cho**(Gyeongsang National University)- Sebastiano Boscarino (University of Catania)
- Giovanni Russo (University of Catania)

**Abstract**: In this work, we propose a conservative semi-Lagrangian method for the Boltzmann equation. Semi-Lagrangian approach enables us to avoid CFL-type restrictions on the time step. High order in time is obtained by Runge-Kutta or Adams-Bashforth methods. High order in space is obtained by a high order conservative reconstruction which also prevents spurious oscillations. The fast spectral method with L2-correction guarantees spectral accuracy and conservation. Numerical results confirm the accuracy and efficiency of the methods.

**[05011] Finite-differences scheme for a tensor PDE model of bionetwork formation and applications****Author(s)**:**Clarissa Astuto**- Giovanni Russo (University of Catania)
- Peter Markowich (King Abdullah University of Science and Technology)
- Daniele Boffi (KAUST)
- Jan Haskovec (KAUST)

**Abstract**: We propose a numerical method for the resolution of a complex biological network. We refer to the Cai-Hu model, where they hypothesized that the topology of the leaf pattern is governed by an optimization of the global energy consumption. The evolution in time of the fluid is governed by an elliptic-parabolic system of partial differential equations and we explore the resulting graph, showing important structural differences when changing the parameters.

**[05021] Efficient implicit methods for the Euler equations in Lagrangian coordinates****Author(s)**:**Simone Chiocchetti**(University of Stuttgart)- Giovanni Russo (University of Catania)
- Sebastiano Boscarino (University of Catania)

**Abstract**: In this talk, we introduce a novel implicit numerical scheme for the multimaterial Euler equations in Lagrangian coordinates. The method takes advantage of the remarkable structure of the governing equations in Lagrangian coordinates, which admits a single scalar wave equation for the pressure field, generating a symmetric positive definite system of linear equations. At the same time, contacts are resolved exactly, due to the Lagrangian nature of the method, even without a Riemann solver.

**[05243] Implicit-explicit time integration for thermal radiative transfer and radiation hydrodynamics****Author(s)**:**Ben Scott Southworth**(Los Alamos National Laboratory)

**Abstract**: Thermal radiative transfer (TRT) is an extremely stiff high-dimensional partial-integro-differential equation, which requires partitioned integration when coupled to hydrodynamics. I introduce an approximation of TRT that captures both stiff asymptotic limits, and IMEX framework requiring only one transport-sweep per timestep. I then discuss nonlinear coupling to hydrodynamics, which is complicated via equation-of-state relations. We derive a temperature closure and framework for semi-implicit-explicit integration of radiation hydrodynamics, demonstrating excellent convergence on stiff radiative shock problems.

**[05310] Carbuncle-free, well-balanced, positivity preserving methods for the shallow water equations, with application to the circular hydraulic jump****Author(s)**:**David I Ketcheson**(King Abdullah University of Science and Technology)

**Abstract**: When a jet of fluid hits a flat plate, the resulting flow consists of two regimes separated by a hydraulic jump. We investigate the behavior of the jump for the shallow water equations. Numerical solvers tend to either exhibit artificial numerical instabilities or suppress the chaotic behavior at high Froude numbers. We propose a new entropy-based Riemann solver that is capable of avoiding carbuncles while allowing the fluid instability to manifest itself.