We propose a novel computational scheme that utilizes a time map, a scalar field representing excitation arrival time, to spatialize the time-dependent dynamics of cardiac and neural signal propagation in multidimensional spaces. The time map can be derived from sequential action potential imaging data, the eikonal equation, or numerical solutions of reaction-diffusion equations governing biological signal propagation. Once computed, the time map can be stored and modified to reflect changes in geometry and domain conductivity. The time map can be used for rapid multidimensional propagation simulations by serving as an instantaneous impulse in systems of ordinary differential equations. Furthermore, a modified time map can be reconstructed from the baseline time map in one-and two-dimensional domains. Numerical experiments and computational simulations are presented to demonstrate the numerical efficiency and effectiveness of the proposed scheme. Practical applications are also explored, such as fast multidimensional simulation with the modified time map and conductivity reconstruction from altered time maps.

A novel mathematical model is proposed to investigate the effects of ephaptic coupling between general neural fiber bundles in a multidimensional space with anisotropic neural fiber bundles. Ephaptic coupling corresponds to the spatiotemporal interaction between propagating fiber bundles through the extracellular space. Adapted from a well-known model in cardiac electrophysiology, the bidomain model comprises of the nonoverlapping intracellular and extracellular space except the common nodes of Ranvier. The proposed two-variable model, a neural bidomain model, is derived from the classic Frankenhaeuser-Huxley model for neural spike propagation along the general neural fiber bundles. The governing equation is mathematically and computationally validated against existing one-dimensional models of neural fiber bundle propagations with aligned nodes of Ranvier. A high-order continuous Galerkin scheme is employed for efficient two-dimensional computational simulation with moving frames, or orthonormal basis vectors, representing the intracellular and extracellular conductivity and nonoverlapping domains. The proposed model is simulated in various two-dimensional configurations of neural fiber bundles that are little known to the community, such as fiber bundles with misaligned nodes of Ranvier, opposite-traveling fiber bundles, and curved fiber bundles. ### Competing Interest Statement The authors have declared no competing interest.

Geometric conduction blocks stop cardiac electric propagation due to the shape or conductivity properties of the domain. The blocks are considered to cause many abnormal cardiac electric propagations, leading to cardiac electrophysiological pathologies, such as cardiac fibrillation and arrhythmia. Locating such multidimensional conduction blocks is challenging, particularly in a complex domain with a complex shape and strong anisotropy, such as the heart. To address this problem, we propose a novel mathematical model of the geometric conduction block using the relative acceleration adopted from space-time physics. An efficient numerical scheme for the mathematical model is also proposed to predict the unidirectional conduction block effectively, even in a complex domain. The relative acceleration in the cardiac electric propagation corresponds to the sink-source relationship between the excited (after repolarization) and excitable (before depolarization) cardiac cells, representing the geometric growth rate of the volume of metric balls. The trajectory is constructed from the wavefront of diffusion-reaction equations by aligning orthonormal basis vectors along the gradient of the action potential. Relative acceleration is computed along the propagational direction from the connection 1-form of the basis vectors. The proposed mathematical model and numerical scheme are applied to demonstrate geometric conduction blocks in two-dimensional (2D) simple curved domains with strong anisotropy.

The covariant derivative is a generalization of differentiating vectors. The Euclidean derivative is a special case of the covariant derivative in Euclidean space. The covariant derivative gathers broad attention, particularly when computing vector derivatives on curved surfaces and volumes in various applications. Covariant derivatives have been computed using the metric tensor from the analytically known curved axes. However, deriving the global axis for the domain has been mathematically and computationally challenging for an arbitrary two-dimensional (2D) surface. Consequently, computing the covariant derivative has been difficult or even impossible. A novel high-order numerical scheme is proposed for computing the covariant derivative on any 2D curved surface. A set of orthonormal vectors, known as moving frames, expand vectors to compute accurately covariant derivatives on 2D curved surfaces. The proposed scheme does not require the construction of curved axes for the metric tensor or the Christoffel symbols. The connectivity given by the Christoffel symbols is equivalently provided by the attitude matrix of orthonormal moving frames. Consequently, the proposed scheme can be extended to the general 2D curved surface. As an application, the Helmholtz‐Hodge decomposition is considered for a realistic atrium and a bunny.

In the numerical discretization of partial differential equations (PDEs) with moving frames on curved surfaces, the discretization error does not converge for a high p≥5. Moreover, the conservation error remains significant even in a refined mesh and does not converge as the polynomial order p increases. We postulate that the inaccurate location of the internal grid points of curved elements causes this problem; this is called the internal point error. This bottleneck of convergence persists even when the vertices of the elements are located exactly on the curved domain. We first theoretically explain the effects of the internal point error on numerical accuracy. We then computationally validate the postulation by using a high-order mesh with a negligible internal point error, even for a high-order polynomial of order p. For computational validations, four differential operators and four PDEs are numerically solved using moving frames on the unit sphere to demonstrate the effect of the internal point error.

Additional grid points are often introduced for the higher-order polynomial of a numerical solution with curvilinear elements. However, those points are likely to be located slightly outside the domain, even when the vertices of the curvilinear elements lie within the curved domain. This misallocation of grid points generates a mesh error, called geometric approximation error . This error is smaller than the discretization error but large enough to significantly degrade a long-time integration. Moreover, this mesh error is considered to be the leading cause of conservation error. Two novel schemes are proposed to improve conservation error and/or discretization error for long-time integration caused by geometric approximation error: The first scheme retrieves the original divergence of the original domain; the second scheme reconstructs the original path of differentiation, called connection , thus retrieving the original connection. The increased accuracies of the proposed schemes are demonstrated by the conservation error for various partial differential equations with moving frames on the sphere.

A novel high-order numerical scheme is proposed to compute the covariant derivative, particularly for divergence and curl, on any curved surface. The proposed scheme does not require the construction of a curved axis or metric tensor, which would deteriorate the accuracy of the covariant derivative and prevent its application to complex surfaces. As an application, the Helmholtz-Hodge decomposition (HHD) is adapted in the context of the Galerkin method for displaying the irrotational, incompressible, and harmonic components of vectors on curved surfaces.

As another critical implementation of moving frames for partial differential equations, this paper proposes a novel numerical scheme by aligning one of three orthogonal unit vectors at each grid point along the direction of a wave propagation to construct an organized set of frames, called a connection. This connection characterizes the geometry of wave propagation depending on (1) the initial point, (2) type of wave, and (3) shape of the domain with conduction properties. The constructed connection is differentiated again to derive the Riemann curvature tensor of orthonormal bases corresponding to important physical and biological meanings in wave propagation. As a practical application, the proposed scheme is applied to diffusion-reaction equations to obtain the Atlas, or a geometric map with connections, of an atrium with cardiac fibers, for the quantitative and qualitative analysis of cardiac action potential propagation, which could contribute to the clinical and surgical planning of atrial fibrillation.

Nektar++ is an open-source framework that provides a flexible, high-performance and scalable platform for the development of solvers for partial differential equations using the high-order spectral/$hp$ element method. In particular, Nektar++ aims to overcome the complex implementation challenges that are often associated with high-order methods, thereby allowing them to be more readily used in a wide range of application areas. In this paper, we present the algorithmic, implementation and application developments associated with our Nektar++ version 5.0 release. We describe some of the key software and performance developments, including our strategies on parallel I/O, on in situ processing, the use of collective operations for exploiting current and emerging hardware, and interfaces to enable multi-solver coupling. Furthermore, we provide details on a newly developed Python interface that enables a more rapid introduction for new users unfamiliar with spectral/$hp$ element methods, C++ and/or Nektar++. This release also incorporates a number of numerical method developments - in particular: the method of moving frames, which provides an additional approach for the simulation of equations on embedded curvilinear manifolds and domains; a means of handling spatially variable polynomial order; and a novel technique for quasi-3D simulations to permit spatially-varying perturbations to the geometry in the homogeneous direction. Finally, we demonstrate the new application-level features provided in this release, namely: a facility for generating high-order curvilinear meshes called NekMesh; a novel new AcousticSolver for aeroacoustic problems; our development of a 'thick' strip model for the modelling of fluid-structure interaction problems in the context of vortex-induced vibrations. We conclude by commenting some directions for future code development and expansion.

When time-dependent partial differential equations (PDEs) are solved numerically in a domain with curved boundary or on a curved surface, mesh error and geometric approximation error caused by the inaccurate location of vertices and other interior grid points, respectively, could be the main source of the inaccuracy and instability of the numerical solutions of PDEs. The role of these geometric errors in deteriorating the stability and particularly the conservation properties are largely unknown, which seems to necessitate very fine meshes especially to remove geometric approximation error. This paper aims to investigate the effect of geometric approximation error by using a high-order mesh with negligible geometric approximation error, even for high order polynomial of order p. To achieve this goal, the high-order mesh generator from CAD geometry called NekMesh is adapted for surface mesh generation in comparison to traditional meshes with non-negligible geometric approximation error. Two types of numerical tests are considered. Firstly, the accuracy of differential operators is compared for various p on a curved element of the sphere. Secondly, by applying the method of moving frames, four different time-dependent PDEs on the sphere are numerically solved to investigate the impact of geometric approximation error on the accuracy and conservation properties of high-order numerical schemes for PDEs on the sphere.

Applying the method of moving frames to Maxwell's equations yields two important advancements for scientific computing. The first is the use of upwind flux for anisotropic materials in Maxwell's equations, especially in the context of discontinuous Galerkin (DG) methods. Upwind flux has been available only to isotropic material, because of the difficulty of satisfying the Rankine–Hugoniot conditions in anisotropic media. The second is to solve numerically Maxwell's equations on curved surfaces without the metric tensor and composite meshes. For numerical validation, spectral convergences are displayed for both two-dimensional anisotropic media and isotropic spheres. In the first application, invisible two-dimensional metamaterial cloaks are simulated with a relatively coarse mesh by both the lossless Drude model and the piecewisely-parametered layered model. In the second application, extremely low frequency propagation on various surfaces such as spheres, irregular surfaces, and non-convex surfaces is demonstrated.

A novel numerical scheme is proposed to solve the shallow water equations (SWEs) on arbitrary rotating curved surfaces. Based on the method of moving frames (MMF) in which the geometry is represented by orthonormal vectors, the proposed scheme not only has the fewest dimensionality both in space and time, but also does not require either of metric tensors, composite meshes, or the ambient space. The MMF–SWE formulation is numerically discretized using the discontinuous Galerkin method of arbitrary polynomial order p in space and an explicit Runge–Kutta scheme in time. The numerical model is validated against six standard tests on the sphere and the optimal order of convergence of p+1 is numerically demonstrated. The MMF–SWE scheme is also demonstrated for its efficiency and stability on the general rotating surfaces such as ellipsoid, irregular, and non-convex surfaces.

Understanding the magnetic field distribution in the heart is crucial for interpreting electrocardiography (ECG) and magnetocardiography (MCG) results. However, its mathematical model has been restricted by the cardiac fibre orientation and has not been successful in some cases. In this paper, a novel electromagnetic model of cardiac electric signal propagation is proposed to provide a complete description of the magnetic field in multidimensional cardiac tissue which can be generated regardless of the cardiac fiber orientation. The derived Maxwell’s equations, which are equivalent to those of the generic two-variable model, are identical to the classical electromagnetic field equations, but for a different medium, and provide a unified macroscopic propagation description of cardiac electric signal in multidimensional anisotropic space.