{
 "cells": [
  {
   "cell_type": "markdown",
   "source": [
    "# 8: Other SBP schemes (FD, CGSEM) via `DGMulti` solver"
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "**Note:** To improve responsiveness via caching, the notebooks are updated only once a week. They are only\n",
    "available for the latest stable release of Trixi.jl at the time of caching."
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "For a tutorial about DG schemes via the `DGMulti` solver please visit the previous tutorial.\n",
    "The `DGMulti` solver also supports other methods than DG. The important property a method has to\n",
    "fulfill is the summation-by-parts (SBP) property. The package [SummationByPartsOperators.jl](https://github.com/ranocha/SummationByPartsOperators.jl)\n",
    "provides such methods, like a finite difference SBP (FD SBP) scheme. To do this,\n",
    "you need to create an SBP derivative operator and pass that as `approximation_type`\n",
    "to the `DGMulti` constructor. For example, the classical second-order FD SBP operator\n",
    "can be created as"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "using Trixi.SummationByPartsOperators # or add SummationByPartsOperators to your project and use it directly\n",
    "D = derivative_operator(MattssonNordström2004(), derivative_order=1, accuracy_order=2,\n",
    "                        xmin=0.0, xmax=1.0, N=11)"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "Here, the arguments `xmin` and `xmax` do not matter beyond setting the real type\n",
    "used for the operator - they just set a reference element and are rescaled on the\n",
    "physical elements. The parameter `N` determines the number of finite difference nodes.\n",
    "Then, `D` can be used as `approximation_type` like `SBP()` in a multi-block fashion.\n",
    "In multiple dimensions, such a 1D SBP operator will be used in a tensor product fashion,\n",
    "i.e., in each coordinate direction. In particular, you can use them only on 1D, 2D `Quad()`,\n",
    "and 3D `Hex()` elements.\n",
    "\n",
    "You can also use fully periodic single-block FD methods by creating a periodic SBP\n",
    "operator. For example, a fully periodic FD operator can be constructed as"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "D = periodic_derivative_operator(derivative_order=1, accuracy_order=2,\n",
    "                                 xmin=0.0, xmax=1.0, N=11)"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "An example using such an FD method is implemented in\n",
    "[`elixir_euler_fdsbp_periodic.jl`](https://github.com/trixi-framework/Trixi.jl/blob/main/examples/dgmulti_2d/elixir_euler_fdsbp_periodic.jl).\n",
    "For all parameters and other calling options, please have a look in the\n",
    "[documentation of SummationByPartsOperators.jl](https://ranocha.de/SummationByPartsOperators.jl/stable/)."
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "Another possible method is for instance a continuous Galerkin (CGSEM) method. You can use such a\n",
    "method with polynomial degree of `3` (`N=4` Legendre Lobatto nodes on `[0, 1]`) coupled continuously\n",
    "on a uniform mesh with `Nx=10` elements by setting `approximation_type` to"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "using Trixi.SummationByPartsOperators # or add SummationByPartsOperators to your project and use it directly\n",
    "D = couple_continuously(legendre_derivative_operator(xmin=0.0, xmax=1.0, N=4),\n",
    "                        UniformPeriodicMesh1D(xmin=-1.0, xmax=1.0, Nx=10))"
   ],
   "metadata": {},
   "execution_count": null
  },
  {
   "cell_type": "markdown",
   "source": [
    "To choose a discontinuous coupling (DGSEM), use `couple_discontinuously()` instead of `couple_continuously()`."
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "For more information and other SBP operators, see the documentations of [StartUpDG.jl](https://jlchan.github.io/StartUpDG.jl/dev/)\n",
    "and [SummationByPartsOperators.jl](https://ranocha.de/SummationByPartsOperators.jl/stable/)."
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Package versions"
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "These results were obtained using the following versions."
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "using InteractiveUtils\n",
    "versioninfo()\n",
    "\n",
    "using Pkg\n",
    "Pkg.status([\"Trixi\", \"StartUpDG\", \"SummationByPartsOperators\"],\n",
    "           mode=PKGMODE_MANIFEST)"
   ],
   "metadata": {},
   "execution_count": null
  }
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