Runge-Kutta Time Integration Research Articles

Hydrodynamic characteristics of blowing and suction on sheet-cavitating flows around hydrofoils

In this study the effect of blowing and suction on hydrodynamic behavior of sheet-cavitating flows over hydrofoils is investigated. A computational program is developed for simulation of inviscid cavitating flows. In this simulation, the Jameson׳s finite volume technique and the progressive power-law preconditioning method for analyzing the cavitating flows are utilized. For cavitation modeling the barotropic cavitation pattern is used. The numerical stabilization is achieved via the second and fourth-order dissipation terms. Explicit four-step Runge–Kutta time integration is applied to achieve the steady-state condition. Cavitating flows over NACA-hydrofoils for different cavitation numbers are investigated. To apply the blowing and suction, a jet is placed on the hydrofoil׳s upper surface. Four blowing (and suction) parameters i.e., velocity ratio, jet location, width of jet, and jet angle are scrutinized. The effects of these parameters on surface pressure distribution, length of cavitation, lift and pressure drag coefficients are examined. The results indicate that blowing jets often decrease the length of cavity and reduce the lift and pressure drag coefficients, while suction jets behave reversely. Also, by increasing the blowing amplitude or the width of jet, the lift and pressure drag coefficients and the length of cavity decrease while in suction cases, they increase.

DYNAMICO-1.0, an icosahedral hydrostatic dynamical core designed for consistency and versatility

Abstract. The design of the icosahedral dynamical core DYNAMICO is presented. DYNAMICO solves the multi-layer rotating shallow-water equations, a compressible variant of the same equivalent to a discretization of the hydrostatic primitive equations in a Lagrangian vertical coordinate, and the primitive equations in a hybrid mass-based vertical coordinate. The common Hamiltonian structure of these sets of equations is exploited to formulate energy-conserving spatial discretizations in a unified way. The horizontal mesh is a quasi-uniform icosahedral C-grid obtained by subdivision of a regular icosahedron. Control volumes for mass, tracers and entropy/potential temperature are the hexagonal cells of the Voronoi mesh to avoid the fast numerical modes of the triangular C-grid. The horizontal discretization is that of Ringler et al. (2010), whose discrete quasi-Hamiltonian structure is identified. The prognostic variables are arranged vertically on a Lorenz grid with all thermodynamical variables collocated with mass. The vertical discretization is obtained from the three-dimensional Hamiltonian formulation. Tracers are transported using a second-order finite-volume scheme with slope limiting for positivity. Explicit Runge–Kutta time integration is used for dynamics, and forward-in-time integration with horizontal/vertical splitting is used for tracers. Most of the model code is common to the three sets of equations solved, making it easier to develop and validate each piece of the model separately. Representative three-dimensional test cases are run and analyzed, showing correctness of the model. The design permits to consider several extensions in the near future, from higher-order transport to more general dynamics, especially deep-atmosphere and non-hydrostatic equations.

Segregated Runge–Kutta methods for the incompressible Navier–Stokes equations

SummaryIn this work, we propose Runge–Kutta time integration schemes for the incompressible Navier–Stokes equations with two salient properties. First, velocity and pressure computations are segregated at the time integration level, without the need to perform additional fractional step techniques that spoil high orders of accuracy. Second, the proposed methods keep the same order of accuracy for both velocities and pressures. The segregated Runge–Kutta methods are motivated as an implicit–explicit Runge–Kutta time integration of the projected Navier–Stokes system onto the discrete divergence‐free space, and its re‐statement in a velocity–pressure setting using a discrete pressure Poisson equation. We have analysed the preservation of the discrete divergence constraint for segregated Runge–Kutta methods and their relation (in their fully explicit version) with existing half‐explicit methods. We have performed a detailed numerical experimentation for a wide set of schemes (from first to third order), including implicit and IMEX integration of viscous and convective terms, for incompressible laminar and turbulent flows. Further, segregated Runge–Kutta schemes with adaptive time stepping are proposed. Copyright © 2015 John Wiley & Sons, Ltd.

Astrophysical hydrodynamics with a high-order discontinuous Galerkin scheme and adaptive mesh refinement

Solving the Euler equations of ideal hydrodynamics as accurately and efficiently as possible is a key requirement in many astrophysical simulations. It is therefore important to continuously advance the numerical methods implemented in current astrophysical codes, especially also in light of evolving computer technology, which favours certain computational approaches over others. Here we introduce the new adaptive mesh refinement (AMR) code TENET, which employs a high order discontinuous Galerkin (DG) scheme for hydrodynamics. The Euler equations in this method are solved in a weak formulation with a polynomial basis by means of explicit Runge-Kutta time integration and Gauss-Legendre quadrature. This approach offers significant advantages over commonly employed second order finite volume (FV) solvers. In particular, the higher order capability renders it computationally more efficient, in the sense that the same precision can be obtained at significantly less computational cost. Also, the DG scheme inherently conserves angular momentum in regions where no limiting takes place, and it typically produces much smaller numerical diffusion and advection errors than a FV approach. A further advantage lies in a more natural handling of AMR refinement boundaries, where a fall-back to first order can be avoided. Finally, DG requires no wide stencils at high order, and offers an improved data locality and a focus on local computations, which is favourable for current and upcoming highly parallel supercomputers. We describe the formulation and implementation details of our new code, and demonstrate its performance and accuracy with a set of two- and three-dimensional test problems. The results confirm that DG schemes have a high potential for astrophysical applications.

Numerical solution of viscous and viscoelastic fluids flow through the branching channel by finite volume scheme

This work deals with the numerical modelling of steady flows of incompressible viscous and viscoelastic fluids through the three dimensional channel with T-junction. The fundamental system of equations is the system of generalized Navier-Stokes equations for incompressible fluids. This system is based on the system of balance laws of mass and momentum for incompressible fluids. Two different mathematical models for the stress tensor are used for simulation of Newtonian and Oldroyd-B fluids flow. Numerical solution of the described models is based on cetral finite volume method using explicit Runge-Kutta time integration.

A third‐order implicit discontinuous Galerkin method based on a Hermite WENO reconstruction for time‐accurate solution of the compressible Navier–Stokes equations

SummaryA space and time third‐order discontinuous Galerkin method based on a Hermite weighted essentially non‐oscillatory reconstruction is presented for the unsteady compressible Euler and Navier–Stokes equations. At each time step, a lower‐upper symmetric Gauss–Seidel preconditioned generalized minimal residual solver is used to solve the systems of linear equations arising from an explicit first stage, single diagonal coefficient, diagonally implicit Runge–Kutta time integration scheme. The performance of the developed method is assessed through a variety of unsteady flow problems. Numerical results indicate that this method is able to deliver the designed third‐order accuracy of convergence in both space and time, while requiring remarkably less storage than the standard third‐order discontinous Galerkin methods, and less computing time than the lower‐order discontinous Galerkin methods to achieve the same level of temporal accuracy for computing unsteady flow problems. Copyright © 2015 John Wiley & Sons, Ltd.

A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension

We implement a high-order numerical scheme for the entropy-based moment closure, the so-called MN model, for linear kinetic equations in slab geometry. A discontinuous Galerkin (DG) scheme in space along with a strong-stability preserving Runge–Kutta time integrator is a natural choice to achieve a third-order scheme, but so far, the challenge for such a scheme in this context is the implementation of a linear scaling limiter when the numerical solution leaves the set of realizable moments (that is, those moments associated with a positive underlying distribution). The difficulty for such a limiter lies in the computation of the intersection of a ray with the set of realizable moments. We avoid this computation by using quadrature to generate a convex polytope which approximates this set. The half-space representation of this polytope is used to compute an approximation of the required intersection straightforwardly, and with this limiter in hand, the rest of the DG scheme is constructed using standard techniques. We consider the resulting numerical scheme on a new manufactured solution and standard benchmark problems for both traditional MN models and the so-called mixed-moment models. The manufactured solution allows us to observe the expected convergence rates and explore the effects of the regularization in the optimization.

On the inter-comparison of two tracer transport schemes on icosahedral grids

Two simple finite volume advection schemes based on linear sub-grid reconstruction are implemented over spherical icosahedral–hexagonal grids. One of theses schemes is fully discrete in space and time, while the other one is a semi-discrete scheme with third order Runge–Kutta time integration. For the linear sub-grid reconstruction, we propose two possible candidates for consistent gradient discretization over general grids. These gradients are designed in a finite volume sense with an adequate modification to guarantee convergence in the absence of special grid optimization. To generate our computational mesh from triangular grid, we either use centroids (BT grid) or circumcenters (VT grid) of the spherical triangular mesh. A numerical convergence study is used to show that for a first order convergence of our discrete gradients grid modification is sufficient, whereas to achieve second order convergence grid optimization is mandatory. This study also implicates that BT grid offers a better rate of convergence than VT grid.Monotonicity of the advection schemes is enforced by a slope limiter as well as flux-corrected transport (FCT). We compared aforementioned space–time coupled and space–time decoupled advection schemes in terms of their performance for the recently proposed advection test cases. Our findings advocate that space–time coupled advection scheme is performing better than its counterpart. Furthermore, we used the fully discrete advection scheme to carried out a comparison of slope limitation and FCT approach to achieve monotonicity.

Fully nonlinear wave interaction with an array of truncated barriers in three dimensional numerical wave tank

Wave transition due to coinciding with an array of truncated barrier is simulated by a fully nonlinear three dimensional potential Numerical Wave Tank (NWT). The potential theory is used to describe kinematics of the flow field and the isoparametric Boundary Element Method (BEM) is employed to solve the boundary value problem. The Mixed Eulerian–Lagrangian (MEL) approach and fourth order Runge–Kutta time integration applied for time-marching scheme to model the temporary and fully nonlinear free surface. At each time step, solution of Laplace equation in the Eulerian frame is applied to the fully nonlinear free surface conditions in the Lagrangian manner to achieve the new positions and the boundary value of fluid particles for the next time step. Normal flux of potential wave theory is specified on the inflow boundary to stimulate fluid field and to propagate the nonlinear wave along the tank. To minimize the reflected wave energy into the computational domain, two artificial sponger layers are adopted on the free surface at the both ends of the numerical wave tank. Accuracy and convergence of the present numerical procedure is conducted. Also, interaction between a near trapped mode array of truncated barriers and nonlinear input wave is simulated.

Implicit integration methods for dislocation dynamics

In dislocation dynamics simulations, strain hardening simulations require integrating stiff systems of ordinary differential equations in time with expensive force calculations, discontinuous topological events and rapidly changing problem size. Current solvers in use often result in small time steps and long simulation times. Faster solvers may help dislocation dynamics simulations accumulate plastic strains at strain rates comparable to experimental observations. This paper investigates the viability of high-order implicit time integrators and robust nonlinear solvers to reduce simulation run times while maintaining the accuracy of the computed solution. In particular, implicit Runge–Kutta time integrators are explored as a way of providing greater accuracy over a larger time step than is typically done with the standard second-order trapezoidal method. In addition, both accelerated fixed point and Newton's method are investigated to provide fast and effective solves for the nonlinear systems that must be resolved within each time step. Results show that integrators of third order are the most effective, while accelerated fixed point and Newton's method both improve solver performance over the standard fixed point method used for the solution of the nonlinear systems.

Aeroelastic analysis and active flutter suppression of an electro-rheological sandwich cylindrical panel under yawed supersonic flow

Supersonic flutter control of a three-layered sandwich curved panel of rectangular plan form with an adaptive electro-rheological fluid (ERF) core layer is investigated. The panel is excited by an arbitrary transient transverse load along with an arbitrary airflow yaw angle. The problem formulation is based on Kirchhoff–Love thin shell theory, the first order Kelvin–Voigt viscoelastic material model, and the linear quasi-steady Krumhaar's modified supersonic piston theory that accounts for both the panel curvature and flow yaw angle. The classical Hamilton's principle and the Galerkin method are used to set up the equations of motion which are then put in the state-space form. The effects of applied electric field, panel aspect ratio, panel shallowness angle, and airflow direction on the critical free stream static pressure are studied. Also, the aeroelastic response of the cylindrical panel excited by an impulsive central point load is calculated using the Runge–Kutta time integration scheme. Subsequently, a Sliding Mode Control (SMC) methodology is implemented to actively attenuate the structural response in the supersonic flow regime. Numerical simulations establish the effectiveness of the applied control configuration in successful suppression of the structural oscillations. The validity of results is demonstrated by rigorous comparisons with the existing data.

The Taylor-SPH Meshfree Method: Basis and Validation

This paper presents the basis and validation of the Taylor-SPH meshless method formulated in terms of stresses and velocities which can be applied to Solid Dynamic problems. The proposed method consists of applying first the time discretization by means of a Taylor series expansion in two steps and a corrected SPH method for the space discretization. In order to avoid numerical instabilities, two different sets of particles are used in the time discretization. To validate the Taylor-SPH method, it has been applied to solve the propagation of shock waves in elastic materials and the results have been compared with those obtained with a corrected SPH discretization combined with a 4th order Runge-Kutta time integration. The Taylor-SPH method is shown to be stable, robust and efficient and it provides more accurate results than those obtained with the standard SPH along with the Runge-Kutta time integration scheme. Numerical dispersion and diffusion are eliminated and only a reduced number of particles is required to obtain accurate results.

An Explicit Nodal Space-Time Discontinuous Galerkin Method for Maxwell's Equations

A novel implicit nodal space-time discontinuous Galerkin (STDG) method is proposed in this letter. An eigenvalue analysis is performed and compared with that for a DG scheme solved with a fourth-order Runge-Kutta time integrator. We show that STDG offers a significant improvement of dissipative and dispersive properties and allows larger time steps, regardless of the spatial hp-refinement. A domain-decomposition technique is used to introduce an explicit formulation of the method in order to render it computationally efficient.

Analysis and application of high order implicit Runge–Kutta schemes to collocated finite volume discretization of the incompressible Navier–Stokes equations

The application of a family of high order implicit Runge–Kutta time integration schemes, namely the explicit first-stage singly diagonally implicit Runge–Kutta schemes (ESDIRK), to cell-centered collocated finite volume discretization of the unsteady incompressible Navier–Stokes is considered. Although achieving computational efficiency relative to commonly used second order implicit schemes has been the motivating factor, this study focuses on temporal order analysis of the high order schemes on the collocated grid. In particular, a face velocity interpolation procedure (Rhie–Chow) which preserves the temporal design order of the ESDIRK schemes is introduced. The details of an iterative pressure-based time advancing algorithm comprising the designed interpolation method are presented (iterated-PISO). The results from solving numerical examples demonstrate the temporal order preservation of the algorithm.

A robust and adaptive recovery‐based discontinuous Galerkin method for the numerical solution of convection–diffusion equations

SUMMARYIn this paper, we introduce and test the enhanced stability recovery (ESR) scheme. It is a robust and compact approach to the computation of diffusive fluxes in the framework of discontinuous Galerkin methods. The scheme is characterized by a new recovery basis and a new procedure for the weak imposition of Dirichlet boundary conditions. These features make the method flexible and robust, even in the presence of highly distorted meshes. The implementation is simplified with respect to the original recovery scheme (RDG1x). Furthermore, thanks to the proposed approach, a robust implementation of p‐adaptive algorithms is possible. Numerical tests on unstructured grids show a convergence rate equal to p + 1, where p is the reconstruction order. Comparisons are shown with the original recovery scheme RDG1x and the widely used BR2 method. Results show a significantly larger stability region for the proposed discretization when explicit Runge–Kutta time integration is employed. Interestingly, this advantage grows quickly when the reconstruction order is increased. The proposed procedure for the weak imposition of Dirichlet boundary conditions does not need the introduction of ghost cells, and it is truly local because it does not require data exchange with other elements. It can be easily used with curvilinear wall elements. Several test cases are considered. They include some benchmark tests with the heat equation and compressible Navier–Stokes equations, with test cases designed also to evaluate the behaviour of the scheme with very stretched elements and separated flows. Copyright © 2014 John Wiley & Sons, Ltd.

Discretely Exact Derivatives for Hyperbolic PDE-Constrained Optimization Problems Discretized by the Discontinuous Galerkin Method

This paper discusses the computation of derivatives for optimization problems governed by linear hyperbolic systems of partial differential equations (PDEs) that are discretized by the discontinuous Galerkin (dG) method. An efficient and accurate computation of these derivatives is important, for instance, in inverse problems and optimal control problems. This computation is usually based on an adjoint PDE system, and the question addressed in this paper is how the discretization of this adjoint system should relate to the dG discretization of the hyperbolic state equation. Adjoint-based derivatives can either be computed before or after discretization; these two options are often referred to as the optimize-then-discretize and discretize-then-optimize approaches. We discuss the relation between these two options for dG discretizations in space and Runge–Kutta time integration. The influence of different dG formulations and of numerical quadrature is discussed. Discretely exact discretizations for several hyperbolic optimization problems are derived, including the advection equation, Maxwell’s equations and the coupled elastic-acoustic wave equation. We find that the discrete adjoint equation inherits a natural dG discretization from the discretization of the state equation and that the expressions for the discretely exact gradient often have to take into account contributions from element faces. For the coupled elastic-acoustic wave equation, the correctness and accuracy of our derivative expressions are illustrated by comparisons with finite difference gradients. The results show that a straightforward discretization of the continuous gradient differs from the discretely exact gradient, and thus is not consistent with the discretized objective. This inconsistency may cause difficulties in the convergence of gradient based algorithms for solving optimization problems.

Perfectly Matched Layers versus discrete transparent boundary conditions in quantum device simulations

Discrete transparent boundary conditions (DTBC) and the Perfectly Matched Layers (PML) method for the realization of open boundary conditions in quantum device simulations are compared, based on the stationary and time-dependent Schrödinger equation. The comparison includes scattering state, wave packet, and transient scattering state simulations in one and two space dimensions. The Schrödinger equation is discretized by a second-order Crank–Nicolson method in case of DTBC. For the discretization with PML, symmetric second-, fourth-, and sixth-order spatial approximations as well as Crank–Nicolson and classical Runge–Kutta time-integration methods are employed. In two space dimensions, a ring-shaped quantum waveguide device is simulated in the stationary and transient regime. As an application, a simulation of the Aharonov–Bohm effect in this device is performed, showing the excitation of bound states localized in the ring region. The numerical experiments show that the results obtained from PML are comparable to those obtained using DTBC, while keeping the high numerical efficiency and flexibility as well as the ease of implementation of the former method.

High-resolution numerical relaxation approximations to second-order macroscopic traffic flow models

A novel numerical approach for the approximation of several, widely applied, macroscopic traffic flow models is presented. A relaxation-type approximation of second-order non-equilibrium models, written in conservation or balance law form, is considered. Using the relaxation approximation, the nonlinear equations are transformed to a semi-linear diagonilizable problem with linear characteristic variables and stiff source terms. To discretize the resulting relaxation system, low- and high-resolution reconstructions in space and implicit–explicit Runge–Kutta time integration schemes are considered. The family of spatial discretizations includes a second-order MUSCL scheme and a fifth-order WENO scheme, and a detailed formulation of the scheme is presented. Emphasis is given on the WENO scheme and its performance for solving the different traffic models. To demonstrate the effectiveness of the proposed approach, extensive numerical tests are performed for the different models. The computations reported here demonstrate the simplicity and versatility of relaxation schemes as solvers for macroscopic traffic flow models.

OFF, Open source Finite volume Fluid dynamics code: A free, high-order solver based on parallel, modular, object-oriented Fortran API

OFF, an open source (free software) code for performing fluid dynamics simulations, is presented. The aim of OFF is to solve, numerically, the unsteady (and steady) compressible Navier–Stokes equations of fluid dynamics by means of finite volume techniques: the research background is mainly focused on high-order (WENO) schemes for multi-fluids, multi-phase flows over complex geometries. To this purpose a highly modular, object-oriented application program interface (API) has been developed. In particular, the concepts of data encapsulation and inheritance available within Fortran language (from standard 2003) have been stressed in order to represent each fluid dynamics “entity” (e.g. the conservative variables of a finite volume, its geometry, etc…) by a single object so that a large variety of computational libraries can be easily (and efficiently) developed upon these objects. The main features of OFF can be summarized as follows:Programming LanguageOFF is written in standard (compliant) Fortran 2003; its design is highly modular in order to enhance simplicity of use and maintenance without compromising the efficiency;Parallel Frameworks Supported the development of OFF has been also targeted to maximize the computational efficiency: the code is designed to run on shared-memory multi-cores workstations and distributed-memory clusters of shared-memory nodes (supercomputers); the code’s parallelization is based on Open Multiprocessing (OpenMP) and Message Passing Interface (MPI) paradigms;Usability, Maintenance and Enhancement in order to improve the usability, maintenance and enhancement of the code also the documentation has been carefully taken into account; the documentation is built upon comprehensive comments placed directly into the source files (no external documentation files needed): these comments are parsed by means of doxygen free software producing high quality html and latex documentation pages; the distributed versioning system referred as git has been adopted in order to facilitate the collaborative maintenance and improvement of the code;CopyrightsOFF is a free software that anyone can use, copy, distribute, study, change and improve under the GNU Public License version 3.The present paper is a manifesto of OFF code and presents the currently implemented features and ongoing developments. This work is focused on the computational techniques adopted and a detailed description of the main API characteristics is reported. OFF capabilities are demonstrated by means of one and two dimensional examples and a three dimensional real application. Program summaryProgram title:OFFCatalogue identifier: AESV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AESV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public Licence, version 3No. of lines in distributed program, including test data, etc.: 60466No. of bytes in distributed program, including test data, etc.: 595575Distribution format: tar.gzProgramming language: Fortran (standard 2003 or newer); developed and tested with Intel Fortran Compiler v. 12.x or newer.Computer: Designed for shared-memory multi-cores workstations and for hybrid distributed/shared-memory supercomputers, but any computer system with a Fortran (2003+) compiler is suited.Operating system: Designed for POSIX architecture and tested on GNU/Linux one.Has the code been vectorized or parallelized?: Hybrid parallelization by means of MPI library and OpenMP paradigm, tested on up to 256 processors.RAM: [1 MB; 1 GB] x core, simulation-dependentClassification: 4.3, 4.10, 12.External routines: The proprietary library [1] must be linked for producing binary outputs in Tecplot Inc. format; the MPI library [2] must be linked for running on distribute-memory parallel systemsNature of problem:Numerical solution of the Compressible Navier–Stokes equations for multi-fluids multi-phase flows in complex geometriesSolution method:Fully-conservative Finite Volume scheme based on very high-order WENO Positivity-Preserving reconstruction technique and Strong Stability Preserving high-order Runge–Kutta time integration. Structured multi-block general curvilinear and body-fitted grids constitute the underlining numerical grids. MPI and OpenMP paradigms are used for parallel computations. Pre-processing tool to deal with commercial meshing softwares is provided as well as post-processing one for numerical results visualizationRestrictions:At present, OFF is validated for simulating inviscid and single-phase flows (viscous fluxes computation and fully coupled Eulerian /Lagrangian schemes have still to be validated, but they are already implemented); modern Fortran compiler is mandatoryUnusual features:OFF is a complex CFD software strongly based on Object-Oriented Programming paradigm by means of modern Fortran standard (2003 or higher)Additional comments:OFF project adopts Git [3], a free and open source distributed version control system. A public repository dedicated to OFF project [4] has been created on github, a web-based hosting service for software development projects using git versioning system. Finally, a comprehensive documentation [5] is provided parsing source code comments by means of doxygen software [6]Running time:The running time depends on the available computation resources, the complexity of the problem and the selected accuracy of the numerical scheme. Simple one dimensional problems require few minutes on personal workstation-like systems whereas complex three dimensional problems needing high accuracy (e.g. DNS) could require weeks on supercomputers. OFF has proven to have a good scalability on small clusters (e.g. CASPUR facilities, namely on Matrix, a GNU/Linux cluster composed by 320 nodes each one constituted by a dual Opteron quadcore at 2.1GHz with 16/32 GB of RAM)

A study of infrasound propagation based on high-order finite difference solutions of the Navier-Stokes equations

The feasibility of using numerical simulation of fluid dynamics equations for the detailed description of long-range infrasound propagation in the atmosphere is investigated. The two dimensional (2D) Navier Stokes equations are solved via high fidelity spatial finite differences and Runge-Kutta time integration, coupled with a shock-capturing filter procedure allowing large amplitudes to be studied. The accuracy of acoustic prediction over long distances with this approach is first assessed in the linear regime thanks to two test cases featuring an acoustic source placed above a reflective ground in a homogeneous and weakly inhomogeneous medium, solved for a range of grid resolutions. An atmospheric model which can account for realistic features affecting acoustic propagation is then described. A 2D study of the effect of source amplitude on signals recorded at ground level at varying distances from the source is carried out. Modifications both in terms of waveforms and arrival times are described.