The social dinner will take place on Wednesday 21st, 8 PM, at "Ristorante Saporito" in via Tiburtina 141.
Here the location on Google Maps
Day 1 - Monday 19th
In this talk we will show a general framework to construct implicit exactly well-balanced finite volume schemes for balance laws with singular source terms. The proposed framework relies on the combination of the Generalized Hydrostatic Reconstruction (GHR) technique and a well-balanced reconstruction operator. First, these two ingredients are recalled as well as the family of semi-discrete in space well-balanced high-order methods obtained by applying them. Then, the extension to implicit time discretizations are discussed. In particular, two new strategies to design exactly well-balanced implicit finite volume schemes for systems with singular source terms will be presented. Finally, the performance of the proposed methods with shown with some numerical tests.
The context of this work is the development of first order total variation diminishing (TVD) implicit-explicit (IMEX) Runge-Kutta (RK) schemes as a basis of a Multidimensional Optimal Order detection (MOOD) approach to approximate the solution of hyperbolic multi-scale equations.
A key feature of our newly proposed TVD schemes is that the resulting CFL condition does not depend on the fast waves of the considered model, as long as they are integrated implicitly.
However, a result from Gottlieb et al. (2001) gives a first order barrier for unconditionally stable implicit TVD-RK schemes and TVD-IMEX-RK schemes with scale-independent CFL conditions.
Therefore, the goal of this work is to consistently improve the resolution of a first-order IMEX-RK scheme, while retaining its $L^\infty$ stability and TVD properties.
In this work we present a novel approach based on a convex combination between a first-order TVD IMEX Euler scheme and a potentially oscillatory high-order IMEX-RK scheme.
We derive and analyse the TVD property for a scalar multi-scale equation and the resulting TVD-MOOD schemes are applied to the isentropic Euler equations.
This is joint work with Victor Michel-Dansac (Inria Strasbourg)
During the past decade, significant effort has been put into high order discontinuous Galerkin methods for compressible flows. A successful design principle has been to find methods that preserve properties of the differential equation, such as the underlying conservation law or the second law of thermodynamics in the form of an entropy inequality.
In this talk we focus on implicit discretizations in time. To solve the arising nonlinear equation systems, iterative methods are needed. The question thus arises if these approximate solutions in turn preserve the properties of the discretization.
Firstly, we consider global and local conservation. As it turns out, many commonly used methods preserve the local conservation of an underlying implicit scheme. This includes pseudo-time iterations, Newton's method and Krylov subspace methods. However, there are prominent exceptions, in particular the Jacobi and Gauss-Seidel iterations. We present extensions of the Lax-Wendroff theorem for a fixed, finite number of iterations each time step. The iterative method defines a numerical flux that is inconsistent in general. We can describe the specific inconsistency as a form of slowed down time for pseudo-time iterations, Krylov methods, and thereby also Newton-Krylov methods.
Secondly, we look at entropy preservation of iterative methods, respectively of nonlinear invariants. As it turns, out, iterative methods do not preserve these. We present various ways to fix this problem by adjusting Newton's method or adding a relaxation step. Numerical methods illustrate the advantages and disadvantages of the various fixes.
Many interesting applications of hyperbolic systems of equations are stiff, in the sense that restrictive CFL conditions are imposed by fields that one is not really interested in tracking accurately. A typical solution in these cases is to resort to implicit time integration, but in the field of high order accurate numerical schemes for hyperbolic equations this is made very difficult by the extreme nonlinearity of the reconstruction operators.
In this talk I will illustrate an approach, that we called Quinpi, to treat nonlinear hyperbolic equations with high order accurate implicit timestepping. It is based on a third order Central WENO reconstruction, a third order DIRK time integrator and a novel idea of time limiting. The scheme is linearized as much as possible using first order accurate predictors for Runge-Kutta stages and the final scheme contains only the nonlinearity of the flux function: in particular it can be applied by solving only linear systems for linear equations. The limiting in time is needed to control spurious oscillations arising from the fact that waves can cross more than one computational cell in each timestep.
Day 2 - Tuesday 20th
Waves in periodic structures have attracted a lot of attention in recent years. Such media show interesting macroscopic properties, which may be quite different from those of the individual fluids which constitute the stratified system. On a macroscopic scale, such a system can be considered a sort of fluid metamaterial. Such waves present a peculiar, somehow unexpected, behavior. For example, there is evidence that in spite of the fact that the waves are governed by genuinely quasi-linear hyperbolic system, they do not break and form shocks.
As specific case, we study the propagation of small amplitude waves in shallow water over a periodic bathymetry [1]. We assume the wavelength is much larger than the period of the bathymetry. It is shown that an initial pulse of small amplitude will not produce a shock. Indeed, after a long time, an initial Gaussian pulse splits into several waves of various amplitude. The phenomenon is not related to the dispersive waves generated from deep water effect, and is explained in terms of dispersive waves satisfying a model system obtained from the original one by suitable asymptotic expansion. The solution to the model system is in good agreement with the detailed numerical solution of the full SW system, agreement that improves with the number of terms in the expansion. Linear stability analysis is adopted to select the more suitable form of the dispersive model. Traveling waves are computed, and compared with the waves that emerge from the detailed dynamics.
[1] David I. Ketcheson, Lajos Loczi, Giovanni Russo, An effective medium equation for weakly nonlinear shallow water waves over periodic bathymetry, MMS, submitted.
The intrinsic diversity in both geometric and mechanical properties of blood vessels contributes to the complex fluid-structure interactions (FSI) that occur between vessel walls and blood flow, posing challenges in the numerical modeling of the human cardiovascular system. In this context, a key objective is to develop a flexible computational model that employs an accurate, efficient, and robust numerical approach.
This talk explores a highly adaptable multiscale constitutive framework designed for modeling one-dimensional blood flow. We demonstrate that diverse blood propagation phenomena can be accurately represented by appropriately selecting scaling parameters linked to different characterizations of the FSI mechanism. To solve the problem numerically, we propose to employ an Asymptotic-Preserving Implicit-Explicit Runge-Kutta Finite Volume method which ensures the consistency of the scheme with various asymptotic limits of the model without compromising its efficiency even when dealing with small scaling parameters.
District heating systems can be modeled by a network of advection equations coupled with a system of differential and algebraic equations. In practical applications, the time interval of interest is large compared to the time steps required by explicit methods. To solve these problems efficiently we investigate high order implicit schemes based on the active flux discretization. Formulating these methods with upwind type stencils, the problem can be solved iteratively. Therefore the numerical effort is comparable to explicit schemes and it can be combined with limiting techniques.
For many mathematical models resulting in stiff problems, only some parts of the given equations are responsible for the stiffness. For instance, in the context of advection-diffusion problems, stiffness is usually induced by the diffusion terms and implicit-explicit (IMEX) time integrators generally discretize the advection terms explicitly and treat the diffusion terms implicitly in order to avoid the corresponding parabolic time step restrictions. Recent research has shown that specific combinations of space and time discretizations even yield unconditional stability, providing grid-independent time step restrictions only based on the advection and diffusion constants. Hereby, implicit discretization of the stiff diffusion terms additionally stabilizes the explicit discretization of the remaining advection terms. This enhanced stability property can be analytically investigated by discrete energy estimates. In this talk, we focus on the spatial discretization by upwind summation-by-parts (SBP) schemes which provide a generic framework to construct robust, structure preserving approximations of higher order and introduce an upwind mechanism similar to numerical fluxes within a discontinuous Galerkin method. In this context, unconditional stability of IMEX upwind-SBP schemes for advection-diffusion problems requires compatibility of the first and second derivative upwind SBP operators. As a second example, we consider kinetic models which describe a wide range of physical processes relevant to natural and engineering sciences where a large number of particles is involved. Collisions of particles are modeled by collision operators which usually lead to stiff problems, in particular for kinetic models in diffusive scaling. In this talk, we will consider specific linear kinetic transport equations modeling radiative transfer. Exploiting the structure-preserving properties of upwind SBP schemes, numerical efficiency can be increased by specific IMEX time integrators, where the micro-macro decomposition of the kinetic equation is suitably split into implicitly and explicity discretized terms. For IMEX upwind SBP schemes, an energy analysis again yields unconditional stability, meaning that arbitrarily large time steps may be chosen near the macroscopic limit of the kinetic model. Specifically, energy stability and error estimates for such schemes will be discussed in this talk.
In this work, we construct and analyze analyze the decay to equilibrium of a finite volume scheme for a one-dimensional kinetic relaxation model describing a generation-recombination reaction of two species. This model is a simplified version of models describing generation and recombination of electron-hole pairs in semiconductor. Using a robust framework originaly developped for studying the large time behavior of continuous linear kinetic equation preserving mass, we prove the exponential decay of our discrete nonlinear model towards its equilibrium distribution.
This is a joint work with M. Bessemoulin-Chatard (Nantes Université) and T. Laidin (Université de Lille).
The goal of this work is the development of advanced numerical approaches for hyperbolic systems with stiff relaxation. In the asymptotic limit, i.e., in the scenario of an infinitesimally small relaxation parameter, denoted as ε, these systems converge to others that may differ in type from the original. Specifically, in the asymptotic limit, a parabolic behavior may emerge, as in the case of the hyperbolic heat equation. As ε approaches zero, the relaxation term becomes very strong and highly stiff, potentially leading to spurious results in numerical simulations.
Implicit-explicit (IMEX) Runge–Kutta schemes have found extensive application in the temporal evolution of hyperbolic partial differential equations (refer to [Pareschi, Russo, J. Sci. Comput., 2005] and [Boscarino, Russo, SIAM J. Sci. Comp., 2009]). Additionally, our objective is to obtain a precisely well-balanced scheme for the asymptotic limit system. This aligns with a technique introduced by two of the authors, which relies on a well-balanced reconstruction operator (detailed in [Castro, Parés, J. Sci. Comput., 2020]). Particular emphasis is placed on accurately approximating the averages and integrals of the source terms through the use of quadrature formulas. We will present numerical experiments to assess the effectiveness of the methods and verify the well-balanced property.
In this talk we will revise the construction of different classes of semi-implicit linear multi-step method for time-dependent PDEs. First of all we will consider the development of high-order space, and time numerical methods based on implicit-explicit multistep time integrators for hyperbolic systems with relaxation. More specifically, we consider hyperbolic balance laws in which the convection and the source term may have very different time and space scales. Hence, we design implicit-explicit linear multistep methods at high-order space-time discretizations which are able to handle all the different scales and to capture the correct asymptotic behavior, independently from its nature, without time step restrictions imposed by the fast scales. Secondly we will consider the construction of semi-implicit linear multistep methods which can be applied to time dependent PDEs where the separation of scales in additive form, is not possible. These semi-implicit techniques give a great flexibility, and allows, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. We will discuss both set-up on different numerical examples, including nonlinear reaction-diffusion and convection-diffusion problems.
Day 3 - Wednesday 21st
Time-dependent multiscale multiphysics partial differential equations (PDEs) are of great practical importance in many fields. Multiscale problems have components evolving at different rates. Multiphysics problems are driven by multiple simultaneous processes with different dynamic characteristics. No single time discretization can solve all components efficiently. In this presentation we discuss a new hybrid time integration framework that allows maximum flexibility in selecting the numerical solution approach for each individual subsystem, while ensuring an overall high order of accuracy.
Arbitrary Derivative (ADER) [Dumbser et al., JCP, 2008] and Deferred Correction (DeC) [Abgrall, JSC, 2017] are arbitrarily high-order methods developed independently in distinct contexts, yet share notable similarities. Both methods employ an iterative process, incrementing by one the order of accuracy at each step. DeC originated as an ODE solver, then used also for more complicated space-time PDE discretizations, while ADER was initially explored as a PDE solver, particularly in its DG space-time discretization, but it has been investigated also as an ODE solver.
Explicit and IMEX versions of these methods have found widespread application, demonstrating adaptability to nonlinear problems through the linearization of stiff terms. We investigate the stability of explicit schemes, revealing a common stability region among these methods for the same order of accuracy [Han Veiga et al., AMC, 2024]. Most implicit methods exhibit A-stability [Petri et al., in prep.], while the IMEX variants are scrutinized by constraining implicit and explicit terms alternately [Minion, Comm. Math. Sci., 2003; Hundsdorfer et al., 2003].
The IMEX formulations of DeC and ADER for PDEs manifest simpler structures in the analysis of linear problems. Our exploration employs von Neumann stability analysis for advection–diffusion and advection–dispersion problems, similarly to [Tan et al. Int. J. Num. Anal. Modeling, 2021]. Hence, we can study the stability regions for some carefully chosen free parameters that depend on time step, mesh discretization and physical coefficients [Petri et al., in prep.]. Our findings establish clear upper bounds for the time step or CFL in advection–diffusion problems. In advection–dispersion cases, the stability regions are much more complicated and show that even respecting similar constraints can lead to unstable schemes.
In a Cartesian embedded boundary approach, a given object is cut out of a Cartesian background mesh, resulting in cut cells along the boundary of the object. These cut cells can be arbitrarily small. When solving time-dependent hyperbolic conservation laws on these meshes, one faces the small cell problem: standard schemes are not stable on the arbitrarily small cut cells if an explicit time stepping scheme is used and the time step size is chosen based on the size of the background cells. A similar issue occurs in the context of two phase flow when sharp interfaces are used for simulating nucleation and cavitation.
In [J. Sci. Comput. 71, 919-943 (2017)], a mixed explicit implicit approach has been introduced to solve the small cell problem: cut cells are treated fully implicitly for stability but away from the cut cells a standard explicit time stepping scheme is used to keep the cost low. In this talk we will explain this approach and discuss how to switch between the explicit and implicit time stepping to ensure conservation of mass. We will further examine how this change affects the accuracy of the scheme [Commun. Appl. Math. Comput., accepted, 2023, arXiv:2310.16101]. We will also discuss applications to two phase flow as presented in [Phys. Fluids 35, 016108 (2023)].
In recent years, there has been a growing interest in modified Patankar schemes due to their ability to maintain the positivity of analytical solutions in production–destruction systems (PDS), regardless of the chosen time step size. These schemes also ensure the conservation property and have found successful applications in biological and chemical systems, as well as serving as effective time-integrators in the semi-discretization of hyperbolic balance laws. However, the stability conditions of modified Patankar schemes and their theoretical properties when applied as time-integrators in hyperbolic balance laws were not clear for a considerable period. The first part of our discussion delves into recent advancements in this field, with a focus on the numerical investigation proposed in [Torlo, Oeffner, Ranocha, App. Numer. Math., 2022], an analytical investigation based on Lyapunov stability from [Izgin, Oeffner, ESAIM: M2AN, 2023], and their interrelationship [Izgin, Oeffner, Torlo, Proceedings of HYP2022, 2022].
Moving to the second part of this presentation, we extend our exploration and showcase the highly efficient application of modified Patankar schemes as a time integration method for hyperbolic balance laws, as highlighted in [Ciallella, Micalizzi, Oeffner, Torlo, Comp. Fluids, 2022]. Towards the conclusion, we present various theoretical properties [Bender, Izgin, Oeffner, Torlo, In preparation, 2024] and initiate a discussion on stability.
This is joint work mainly with D. Torlo (Sissa) and T. Izgin (Uni Kassel). Further contributions are done with J. Bender (Uni Kassel), H. Ranocha (Johannes-Gutenberg University Mainz), M. Ciallella (University Bordeaux) and L. Micalizzi (University Zuerich).
Day 4 - Thursday 22nd
We present a discontinuity aware quadrature (DAQ) rule, and use it to develop implicit self-adaptive theta (SATh) schemes for the approximation of scalar hyperbolic conservation laws. Our SATh schemes require the solution of a system of two equations, one controlling the cell averages of the solution at the time levels, and the other controlling the space-time averages of the solution. These quantities are used within the DAQ rule to approximate the time integral of the hyperbolic flux function accurately, even when the solution may be discontinuous somewhere over the time interval. We prove that DAQ is accurate to second order when there is a discontinuity in the solution and third order when it is smooth. The resulting scheme is a finite volume theta time stepping method, with theta defined implicitly (or self-adaptively). When upstream weighted, SATh-up is unconditionally stable, satisfies the maximum principle, and is total variation diminishing under appropriate monotonicity and boundary conditions, provided that theta is set to be at least 1/2. The scheme can be reformulated so that it is L-Stable for the linear problem, provided only that theta is set to be at least zero. We present numerical results showing the performance of the SATh schemes, sometimes using the more general Lax-Friedrichs numerical flux. Compared to solutions of finite volume schemes using Crank-Nicolson and backward Euler time stepping, SATh solutions often approach the accuracy of the former but without oscillation, and they are numerically less diffuse than the later.
We consider high-order Diagonally Implicit Runge-Kutta time integration methods coupled with Discontinuous Galerkin space reconstructions for conservation laws. We analyze the dispersion and diffusion properties for different pairs of schemes to select the best combination for high Courant numbers in terms of controlling spurious oscillations. Working with high-order methods, however, it is necessary to introduce local slope limiters. We therefore propose to control numerical oscillations by limiting only the high-order terms of the DG reconstruction, using appropriate weights based on a first-order predictor.
We consider equations of motion associated with the semi-convection problem in the simulation of stellar convection, and some other models of flow with diffusion that appear in astrophysics.
These problems are firstly semi-discretized in space by essentially non-oscillatory schemes and dissipative finite difference methods, and they are subsequently integrated in time by IMplicit-EXplicit (IMEX) Runge-Kutta methods which are constructed to preserve different stability properties. [1, 2]. For some simple examples, as well as for the problem of double-diffusive convection, it can be demonstrated that they provide a significant computational advantage over other methods from the literature.
Ongoing work continues with the study of some implementation issues as well as with the study and construction of robust IMEX schemes for some other problems in astrophysics.
This is a joint work with Othmar Koch, Friedrich Kupka and Teo Roldàn.
[1] F. Kupka, N. Happenhofer, I. Higueras, O. Koch. Total-variation-diminishing implicit-explicit Runge-Kutta methods for the simulation of double-diffusive convection in astrophysics, J. Comput. Phys. 232 (2012), 3561-3586.
[2] I. Higueras, N. Happenhofer, O. Koch, F. Kupka. Optimized strong stability preserving IMEX Runge-Kutta methods, J. Comput. Appl. Math. 272 (2014), 116-140.
The efficient numerical solution of compressible Euler equations of gas dynamics poses several computational challenges. Severe time restrictions are required by standard explicit time discretization techniques, in particular for flow regimes characterized by low Mach number values. We propose an efficient, accurate, and robust solver for Euler equations. In addition to the results presented in previous works, we prove the asymptotic-preserving (AP) property of the numerical scheme in the low Mach number limit. The analysis is carried out considering a general equation of state (EOS) and therefore it is not restricted to the ideal gas law as done for standard AP schemes. The method is based on an Additive Runge Kutta IMEX (ARK-IMEX) method for time discretization and allows to simulate low Mach number regimes at a reduced computational cost, while maintaining full accuracy also in case of higher Mach number regimes. We couple implicitly the energy equation to the momentum one, while treating the continuity equation in an explicit fashion. The scheme has been implemented in the framework of the deal.II numerical library, using an h-adaptive Discontinuous Galerkin spatial discretization, so that adaptive mesh refinement capabilities are employed to enhance efficiency. The method is effective also for real gases equations of state such as the general cubic EOS or the stiffened gas equation of state (SG-EOS). A number of numerical experiments on classical benchmarks for compressible flows, in particular applications for atmosphere dynamics, and their extension to real gases demonstrate the properties of the proposed scheme.
Joint work with Luca Bonaventura (Politecnico di Milano).
We have recently focused our attention on using general linear methods (GLMs) as a framework to analyze and generalize existing classes of numerical methods for ordinary differential equations. In particular, starting from Runge-Kutta methods, this approach can lead to the definition of Self Starting GLMs that are multi-stage multi-step methods, which do not require any additional starting procedure. We showed how some properties of Runge-Kutta methods can be improved, keeping similar computational costs. This analysis indicates that the proposed methods may have better accuracy and stability properties, such as, for example, larger stability regions in the case of explicit methods, or stage order greater than one for singly diagonally implicit methods. The possibility to identify good families of implicit and explicit methods with a larger number of free parameters allows the determination of new efficient and highly stable Implicit-Explicit (IMEX) methods.
In this talk, after an overview on general linear methods, we show how to introduce the class of Self Starting Implicit-Explicit General Linear Methods, showing how the extra free parameters can also be exploited to get some additional properties such as Asyptotic Preserving (AP) and Asymptotic Accuracy (AA) for relaxation problems under several conditions on the coefficients of the method as, for example, the Globally Stiffly Accuracy (GSA).
Finally, we report numerical experiments which confirm that Self Starting IMEX GLMs are competitive and can have better performance than IMEX Runge-Kutta methods.
The aim of this work is to apply a semi-implicit (SI) strategy in an implicit-explicit (IMEX) Runge–Kutta (RK) setting introduced in Boscarino et al. (J Sci Comput 68:975–1001, 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. Delta t = O(\Delta t^k) for the k-th (k >= 2) order PDEs. For the space discretization, this strategy is combined with finite difference 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.
In this presentation, we show recent advances on the use of IMEX multiderivative methods for both ODEs and PDEs.
Multiderivative methods turned out to perform very efficiently for, e.g., applications in computational fluid dynamics. In contrast to more 'classical' time integration schemes, not only the ODE's flux, but also its temporal derivative is taken into account. There is a certain peculiarity when it comes to IMEX multiderivative schemes: Even if the implicit part is linear, the second derivative of this implicit part, needed by many algorithms, is nonlinear if the explicit part is. Obviously, this renders the algorithm rather inefficient. We show how this can be alleviated in a spectrally deferred correction setting.
Numerical results covering stiff ODEs and PDEs are shown. Also the asymptotic preserving property for low-Mach Euler equation is briefly touched upon.
Day 5 - Friday 23rd
This talk presents a family of algebraically constrained finite element schemes for nonlinear hyperbolic problems. The validity of (generalized) discrete maximum principles is enforced using monolithic convex limiting, a new flux correction procedure based on representation of spatial semi-discretizations in terms of admissible intermediate states. Semi-discrete entropy stability is enforced using a limiter-based fix. Extensions to high-order (Bernstein or Legendre-Gauss-Lobatto, continuous or discontinuous) finite element approximations assemble algebraic stabilization terms from subcell fluxes in order to minimize the levels of artificial viscosity and the computational cost. Time integration is performed using explicit or implicit Runge-Kutta methods, which can also be equipped with property-preserving flux limiters. In extensions to nonlinear systems, problem-dependent inequality constraints are imposed on scalar functions of conserved variables to ensure physical and numerical admissibility. After explaining the design philosophy behind such high-resolution finite element schemes, we analyze their properties and show some numerical examples.
Runge-Kutta (RK) methods may demonstrate order reduction when applied to stiff problems. This talk explores the issue of order reduction in Runge-Kutta methods specifically when dealing with linear and semilinear stiff problems. First, I will introduce Diagonally Implicit Runge-Kutta (DIRK) methods with high Weak Stage Order (WSO), capable of mitigating order reduction in linear problems with time-independent operators. Following that, I will discuss explicit RK methods with high WSO, tailored for the initial-boundary value problem with time-dependent boundary conditions in hyperbolic fields. On the theoretical front, I will present order barriers relating the WSO of an RK scheme to its order and the number of stages for fully-implicit RK, DIRK, and ERK schemes, serving as a foundation to construct schemes with high WSO. Lastly, I will conclude by presenting stiff order conditions for semilinear problems, essential to extend beyond the limitations of WSO, which primarily focused on linear problems.
Semi-Lagrangian schemes are characteristic-based schemes commonly employed to treat advection terms in time-dependent PDEs. They preserve stability under large Courant numbers, hence they may be appealing in many practical situations. However, the need of locating the feet of characteristics may cause a serious drop of efficiency in the case of unstructured space grids.
In this talk, I will present an in-depth analysis of the main recipes available for characteristic location on unstructured grids, and propose a simple but effective technique to speed-up this phase of the computation, exploiting additional information related to the advecting vector field of the underlying PDE.
Finally, I will show the results obtained applying the proposed method to different optimal navigation problems, both in the case of stationary and non stationary drift fields.
Work in collaboration with R. Ferretti and G. Tatafiore
In this talk, we present a class of higher-order [1] and high-resolution [2] compact implicit or semi-implicit methods for some representative hyperbolic partial differential equations used in conservation laws or in level set methods. Our motivation is similar to fully implicit time discretization methods for conservation laws in [3,4], that is, to offer unconditionally stable numerical schemes. Opposite to the quoted methods where the spatial and temporal discretizations are applied separately, we couple the both discretizations to obtain algebraic systems that are much easier to solve than in the case of fully implicit schemes.
The methods can be formally derived using a partial Lax-Wendroff (or Cauchy-Kowalevski) procedure where the time derivatives in the Taylor series are replaced by the mixed derivatives exploiting the partial differential equation. This is a main modification of the standard L-W procedure in which the time derivatives are replaced by spatial derivatives only. Using some appropriate approximations of the mixed derivatives, one obtains a convenient form of the Jacobian for the resulting nonlinear algebraic equations (or the matrix in the case of linear equations) when the system of equations can be solved efficiently using, for example, a fast sweeping strategy [5].
The details will be given for nonlinear nonconservative advection equations in two-dimensional case for which unconditionally stable third order accurate method is derived [1] and for nonlinear conservation laws in one-dimensional case for which TVD scheme is derived in [2].
[1] Frolkovič, P., & Gajdošová, N. (2024). Unconditionally stable higher order semi-implicit level set method for advection equations. Applied Mathematics and Computation, 466.
[2] Frolkovič, P., & Žeravý, M. (2023). High resolution compact implicit numerical scheme for conservation laws. Applied Mathematics and Computation, 442.
[3] Arbogast, Todd, et al. (2020). A third order, implicit, finite volume, adaptive Runge–Kutta WENO scheme for advection–diffusion equations." Computer Methods in Applied Mechanics and Engineering 368.
[4] G. Puppo, M. Semplice, G. Visconti (2022). Quinpi: integrating conservation laws with CWENO implicit methods. Commun. Appl. Math. Comput.
[5] E. Lozano, T. D. Aslam (2021). Implicit fast sweeping method for hyperbolic systems of conservation laws. J. Comput. Phys., 430.