Table of Content

20 March 2025, Volume 7 Issue 1

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Preface to Focused Section on Efficient High-Order Time Discretization Methods for Partial Differential Equations

Sebastiano Boscarino, Giuseppe Izzo, Lorenzo Pareschi, Giovanni Russo, Chi-Wang Shu

2025, 7(1):  1-2.  doi:10.1007/s42967-024-00464-0

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On Error-Based Step Size Control for Discontinuous Galerkin Methods for Compressible Fluid Dynamics

Hendrik Ranocha, Andrew R. Winters, Hugo Guillermo Castro, Lisandro Dalcin, Michael Schlottke-Lakemper, Gregor J. Gassner, Matteo Parsani

2025, 7(1):  3-39.  doi:10.1007/s42967-023-00264-y

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We study a temporal step size control of explicit Runge-Kutta (RK) methods for compressible computational fluid dynamics (CFD), including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations. We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy (CFL) number. Our numerical examples show that the error-based step size control is easy to use, robust, and efficient, e.g., for (initial) transient periods, complex geometries, nonlinear shock capturing approaches, and schemes that use nonlinear entropy projections. We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases, the open source Julia packages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc.

Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order

Lorenzo Micalizzi, Davide Torlo, Walter Boscheri

2025, 7(1):  40-77.  doi:10.1007/s42967-023-00290-w

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We propose a new paradigm for designing efficient p-adaptive arbitrary high-order methods. We consider arbitrary high-order iterative schemes that gain one order of accuracy at each iteration and we modify them to match the accuracy achieved in a specific iteration with the discretization accuracy of the same iteration. Apart from the computational advantage, the newly modified methods allow to naturally perform the p-adaptivity, stopping the iterations when appropriate conditions are met. Moreover, the modification is very easy to be included in an existing implementation of an arbitrary high-order iterative scheme and it does not ruin the possibility of parallelization, if this was achievable by the original method. An application to the Arbitrary DERivative (ADER) method for hyperbolic Partial Differential Equations (PDEs) is presented here. We explain how such a framework can be interpreted as an arbitrary high-order iterative scheme, by recasting it as a Deferred Correction (DeC) method, and how to easily modify it to obtain a more efficient formulation, in which a local a posteriori limiter can be naturally integrated leading to the p-adaptivity and structure-preserving properties. Finally, the novel approach is extensively tested against classical benchmarks for compressible gas dynamics to show the robustness and the computational efficiency.

Asymmetry and Condition Number of an Elliptic-Parabolic System for Biological Network Formation

Clarissa Astuto, Daniele Boffi, Jan Haskovec, Peter Markowich, Giovanni Russo

2025, 7(1):  78-94.  doi:10.1007/s42967-023-00297-3

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We present results of numerical simulations of the tensor-valued elliptic-parabolic PDE model for biological network formation. The numerical method is based on a nonlinear finite difference scheme on a uniform Cartesian grid in a two-dimensional (2D) domain. The focus is on the impact of different discretization methods and choices of regularization parameters on the symmetry of the numerical solution. In particular, we show that using the symmetric alternating direction implicit (ADI) method for time discretization helps preserve the symmetry of the solution, compared to the (non-symmetric) ADI method. Moreover, we study the effect of the regularization by the isotropic background permeability r > 0, showing that the increased condition number of the elliptic problem due to decreasing value of r leads to loss of symmetry. We show that in this case, neither the use of the symmetric ADI method preserves the symmetry of the solution. Finally, we perform the numerical error analysis of our method making use of the Wasserstein distance.

Beyond Strang: a Practical Assessment of Some Second-Order 3-Splitting Methods

Raymond J. Spiteri, Arash Tavassoli, Siqi Wei, Andrei Smolyakov

2025, 7(1):  95-114.  doi:10.1007/s42967-023-00314-5

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Operator-splitting is a popular divide-and-conquer strategy for solving differential equations. Typically, the right-hand side of the differential equation is split into a number of parts that are then integrated separately. Many methods are known that split the righthand side into two parts. This approach is limiting, however, and there are situations when 3-splitting is more natural and ultimately more advantageous. The second-order Strang operator-splitting method readily generalizes to a right-hand side splitting into any number of operators. It is arguably the most popular method for 3-splitting because of its efficiency, ease of implementation, and intuitive nature. Other 3-splitting methods exist, but they are less well known, and analysis and evaluation of their performance in practice are scarce. We demonstrate the effectiveness of some alternative 3-splitting, second-order methods to Strang splitting on two problems: the reaction-diffusion Brusselator, which can be split into three parts that each have closed-form solutions, and the kinetic Vlasov-Poisson equations that are used in semi-Lagrangian plasma simulations. We find alternative second-order 3-operator-splitting methods that realize efficiency gains of 10%–20% over traditional Strang splitting. Our analysis for the practical assessment of the efficiency of operator-splitting methods includes the computational cost of the integrators and can be used in method design.

The High-Order Variable-Coefficient Explicit-Implicit-Null Method for Diffusion and Dispersion Equations

Meiqi Tan, Juan Cheng, Chi-Wang Shu

2025, 7(1):  115-150.  doi:10.1007/s42967-023-00359-6

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For the high-order diffusion and dispersion equations, the general practice of the explicitimplicit-null (EIN) method is to add and subtract an appropriately large linear highest derivative term with a constant coefficient at one side of the equation, and then apply the standard implicit-explicit method to the equivalent equation. We call this approach the constant-coefficient EIN method in this paper and hereafter denote it by “CC-EIN”. To reduce the error in the CC-EIN method, the variable-coefficient explicit-implicit-null (VC-EIN) method, which is obtained by adding and subtracting a linear highest derivative term with a variable coefficient, is proposed and studied in this paper. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the VC-EIN method is shown to be unconditionally stable and can achieve high order of accuracy for both one-dimensional and twodimensional quasi-linear and nonlinear equations. In addition, although the computational cost slightly increases, the VC-EIN method can obtain more accurate results than the CCEIN method, if the diffusion coefficient or the dispersion coefficient has a few high and narrow bumps and the bumps only account for a small part of the whole computational domain.

Semi-implicit-Type Order-Adaptive CAT2 Schemes for Systems of Balance Laws with Relaxed Source Term

Emanuele Macca, Sebastiano Boscarino

2025, 7(1):  151-178.  doi:10.1007/s42967-024-00414-w

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In this paper, we present two semi-implicit-type second-order compact approximate Taylor (CAT2) numerical schemes and blend them with a local a posteriori multi-dimensional optimal order detection (MOOD) paradigm to solve hyperbolic systems of balance laws with relaxed source terms. The resulting scheme presents the high accuracy when applied to smooth solutions, essentially non-oscillatory behavior for irregular ones, and offers a nearly fail-safe property in terms of ensuring the positivity. The numerical results obtained from a variety of test cases, including smooth and non-smooth well-prepared and unprepared initial conditions, assessing the appropriate behavior of the semi-implicit-type second order CATMOOD schemes. These results have been compared in the accuracy and the efficiency with a second-order semi-implicit Runge-Kutta (RK) method.

Characterizations and Properties of Dual Matrix Star Orders

Hongxing Wang, Pei Huang

2025, 7(1):  179-202.  doi:10.1007/s42967-023-00255-z

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In this paper, we introduce the D-star order, T-star order, and P-star order on the class of dual matrices. By applying the matrix decomposition and dual generalized inverses, we discuss properties, characterizations, and relations among these orders, and illustrate their relations with examples.

L1/LDG Method for Caputo-Hadamard Time Fractional Diffusion Equation

Zhen Wang

2025, 7(1):  203-227.  doi:10.1007/s42967-023-00257-x

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In this paper, a class of discrete Gronwall inequalities is proposed. It is efficiently applied to analyzing the constructed L1/local discontinuous Galerkin (LDG) finite element methods which are used for numerically solving the Caputo-Hadamard time fractional diffusion equation. The derived numerical methods are shown to be α-robust using the newly established Gronwall inequalities, that is, it remains valid when α → 1^-. Numerical experiments are given to demonstrate the theoretical statements.

Motion, Dual Quaternion Optimization and Motion Optimization

Liqun Qi

2025, 7(1):  228-238.  doi:10.1007/s42967-023-00262-0

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We regard a dual quaternion as a real eight-dimensional vector and present a dual quaternion optimization model. Then we introduce motions as real six-dimensional vectors. A motion means a rotation and a translation. We define a motion operator which maps unit dual quaternions to motions, and a UDQ operator which maps motions to unit dual quaternions. By these operators, we present another formulation of dual quaternion optimization. The objective functions of such dual quaternion optimization models are real valued. They are different from the previous model whose object function is dual number valued. This avoids the two-stage problem, which causes troubles sometimes. We further present an alternative formulation, called motion optimization, which is actually an unconstrained real optimization model. Then we formulate two classical problems in robot research, i.e., the hand-eye calibration problem and the simultaneous localization and mapping (SLAM) problem as such dual quaternion optimization problems as well as such motion optimization problems. This opens a new way to solve these problems.

Discovery of Governing Equations with Recursive Deep Neural Networks

Jarrod Mau, Jia Zhao

2025, 7(1):  239-263.  doi:10.1007/s42967-023-00270-0

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Model discovery based on existing data has been one of the major focuses of mathematical modelers for decades. Despite tremendous achievements in model identification from adequate data, how to unravel the models from limited data is less resolved. This paper focuses on the model discovery problem when the data is not efficiently sampled in time, which is common due to limited experimental accessibility and labor/resource constraints. Specifically, we introduce a recursive deep neural network (RDNN) for data-driven model discovery. This recursive approach can retrieve the governing equation efficiently and significantly improve the approximation accuracy by increasing the recursive stages. In particular, our proposed approach shows superior power when the existing data are sampled with a large time lag, from which the traditional approach might not be able to recover the model well. Several examples of dynamical systems are used to benchmark this newly proposed recursive approach. Numerical comparisons confirm the effectiveness of this recursive neural network for model discovery. The accompanying codes are available at https:// github. com/ c2fd/ RDNNs.

Optimal Error Analysis of Linearized Crank-Nicolson Finite Element Scheme for the Time-Dependent Penetrative Convection Problem

Min Cao, Yuan Li

2025, 7(1):  264-288.  doi:10.1007/s42967-023-00269-7

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This paper focuses on the optimal error analysis of a linearized Crank-Nicolson finite element scheme for the time-dependent penetrative convection problem, where the mini element and piecewise linear finite element are used to approximate the velocity field, the pressure, and the temperature, respectively. We proved that the proposed finite element scheme is unconditionally stable and the optimal error estimates in L²-norm are derived. Finally, numerical results are presented to confirm the theoretical analysis.

A System of Hamilton-Jacobi Equations Characterizing Geodesic Centroidal Tessellations

Fabio Camilli, Adriano Festa

2025, 7(1):  289-314.  doi:10.1007/s42967-023-00276-8

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We introduce a class of systems of Hamilton-Jacobi equations characterizing geodesic centroidal tessellations, i.e., tessellations of domains with respect to geodesic distances where generators and centroids coincide. Typical examples are given by geodesic centroidal Voronoi tessellations and geodesic centroidal power diagrams. An appropriate version of the Fast Marching method on unstructured grids allows computing the solution of the Hamilton-Jacobi system and, therefore, the associated tessellations. We propose various numerical examples to illustrate the features of the technique.

Numerical Approach for Solving Two-Dimensional Time-Fractional Fisher Equation via HABC-N Method

Ren Liu, Lifei Wu

2025, 7(1):  315-346.  doi:10.1007/s42967-023-00282-w

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For the two-dimensional time-fractional Fisher equation (2D-TFFE), a hybrid alternating band Crank-Nicolson (HABC-N) method based on the parallel finite difference technique is proposed. The explicit difference method, implicit difference method, and C-N difference method are used simultaneously with the alternating band technique to create the HABC-N method. The existence of the solution and unconditional stability for the HABC-N method, as well as its uniqueness, are demonstrated by theoretical study. The HABC-N method’s convergence order is O(τ^2-α + h₁² + h₂²). The theoretical study is bolstered by numerical experiments, which establish that the 2D-TFFE can be solved using the HABC-N method.

The Hermite-Taylor Correction Function Method for Maxwell’s Equations

Yann-Meing Law, Daniel Appel?

2025, 7(1):  347-371.  doi:10.1007/s42967-023-00287-5

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The Hermite-Taylor method, introduced in 2005 by Goodrich et al. is highly efficient and accurate when applied to linear hyperbolic systems on periodic domains. Unfortunately, its widespread use has been prevented by the lack of a systematic approach to implementing boundary conditions. In this paper we present the Hermite-Taylor correction function method (CFM), which provides exactly such a systematic approach for handling boundary conditions. Here we focus on Maxwell’s equations but note that the method is easily extended to other hyperbolic problems.

On the Order of Accuracy of Edge-Based Schemes: a Peterson-Type Counter-Example

Pavel Bakhvalov, Mikhail Surnachev

2025, 7(1):  372-391.  doi:10.1007/s42967-023-00292-8

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Numerical schemes for the transport equation on unstructured meshes usually exhibit the convergence rate p ∈ [k, k + 1], where k is the order of the truncation error. For the discontinuous Galerk in method, the result p = k + 1∕2 is known, and the example where the convergence rate is exactly k + 1∕2 was constructed by Peterson (SIAM J. Numer. Anal. 28: 133–140, 1991) for k = 0 and k = 1. For finite-volume methods with k≥1, there are no theoretical results for general meshes. In this paper, we consider three edge-based finitevolume schemes with k = 1, namely the Barth scheme, the Luo scheme, and the EBR3. For a special family of meshes, under stability assumption we prove the convergence rate p = 3∕2 for the Barth scheme and p = 5∕4 for the other ones. We also present a Petersontype example showing that the values 3∕2 and 5∕4 are optimal.

An Order Reduction Method for the Nonlinear Caputo-Hadamard Fractional Diffusion-Wave Model

Jieying Zhang, Caixia Ou, Zhibo Wang, Seakweng Vong

2025, 7(1):  392-408.  doi:10.1007/s42967-023-00295-5

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In this paper, the numerical solutions of the nonlinear Hadamard fractional diffusion-wave model with the initial singularity are investigated. Firstly, the model is transformed into coupled equations by virtue of a symmetric fractional-order reduction method. Then the L_log,2-1_σ formula on nonuniform grids is applied to approach to the time fractional derivative. In addition, the discrete fractional Grönwall inequality is used to analyze the optimal convergence of the constructed numerical scheme by the energy method. The accuracy of the theoretical analysis will be demonstrated by means of a numerical experiment at the end.