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 1 gross 3379 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % 4 % Copyright (c) 2003-2010 by University of Queensland 5 % Earth Systems Science Computational Center (ESSCC) 6 7 % 8 % Primary Business: Queensland, Australia 9 % Licensed under the Open Software License version 3.0 10 11 % 12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13 14 15 \section{Some Remarks on Lumping} 16 \label{REMARKS ON LUMPING} 17 18 ahallam 3381 Explicit time integration schemes (two examples are discussed later in this 19 section), require very small time steps in order to maintain numerical stability. 20 Unfortunately, these small time increments often result in a prohibitive 21 computational cost. 22 In order to minimise these costs, a technique termed lumping can be utilised. 23 Lumping is applied to the coefficient matrix, reducing it to a simple diagonal 24 matrix. This can significantly improve the computational speed, because the 25 solution updates are simplified to a simple component-by-component 26 vector-vector product. However, some care is required when making radical 27 approximations such as these. In this section, two commonly applied lumping 28 techniques are discussed, namely row sum lumping 29 gross 3379 \index{linear solver!row sum lumping}\index{row sum lumping} and HRZ 30 lumping \index{linear solver!HRZ lumping}\index{HRZ lumping}. 31 32 ahallam 3381 \subsection{Scalar wave equation} 33 One example where lumping can be applied to a hyperpolic problem, is 34 the scalar wave equation 35 gross 3379 \begin{eqnarray} \label{LUMPING WAVE} 36 u_{,tt}=c^2 u_{,ii} \; . 37 \end{eqnarray} 38 ahallam 3381 In this example, both of the lumping schemes are tested against the reference solution 39 gross 3379 \begin{eqnarray} \label{LUMPING WAVE TEST} 40 u=sin(5 \pi (x_0-c*t) ) 41 \end{eqnarray} 42 over the 2D unit square. Note that $u_{,i}n_i=0$ on faces $x_1=0$ and $x_1=1$. 43 ahallam 3381 Thus, on the faces $x_0=0$ and $x_0=1$ the solution is constrained. 44 gross 3379 45 ahallam 3381 To solve this problem the explicit Verlet scheme\index{Verlet scheme} was used 46 with a constant time step size $dt$ given by 47 gross 3379 \begin{eqnarray} \label{LUMPING WAVE VALET} 48 ahallam 3381 u^{(n)}=2u^{(n-1)}-u^{(n-2)} + dt^2 a^{(n)} 49 gross 3379 \end{eqnarray} 50 ahallam 3381 for all $n=2,3,\ldots$ where the upper index ${(n)}$ refers to values at 51 time $t^{(n)}=t^{(n-1)}+h$ and $a^{(n)}$ is the solution of 52 gross 3379 \begin{eqnarray} \label{LUMPING WAVE VALET 2} 53 a^{(n)}=c^2 u^{(n-1)}_{,ii} \; . 54 \end{eqnarray} 55 ahallam 3381 This equation can be interpreted as a PDE for the unknown value $a^{(n)}$, 56 which must be solved at each time-step. 57 In the notation of equation~\ref{LINEARPDE.SINGLE.1} we thus set $D=1$ and 58 $X=-c^2 u^{(n-1)}_{,i}$. Furthermore, in order to maintain stability, 59 the time step size needs to fullfill the Courant–Friedrichs–Lewy condition 60 (CFL condition). 61 \index{Courant condition} 62 \index{explicit scheme!Courant condition} 63 For this example, the CFL condition takes the form 64 gross 3379 \begin{eqnarray} \label{LUMPING WAVE CFL} 65 dt = f \cdot \frac{dx}{c} . 66 \end{eqnarray} 67 ahallam 3381 where $dx$ is the mesh size and $f$ is a safety factor. In this example, 68 we use $f=\frac{1}{6}$. 69 gross 3379 70 ahallam 3381 Figure~\ref{FIG LUMPING VALET A} depicts a temporal comparison between four 71 alternative solution algorithms: the exact solution; using a full mass matrix; 72 using HRZ lumping; and row sum lumping. The domain utilsed rectangular order 1 73 elements (element size is $0.01$) with observations taken at the point 74 $(\frac{1}{2},\frac{1}{2})$. 75 All four solutions appear to be identical for this example. This is not the case 76 for order $2$ elements, as illustrated in Figure~\ref{FIG LUMPING VALET B}. 77 For the order $2$ elements, the row sum lumping has become unstable. Row sum 78 lumping is unstable in this case because for order $2$ elements, a row sum can 79 result in a value of zero. HRZ lumping does not display the same problems, but 80 rather exhibits behaviour similar to the full mass matrix solution. When using 81 both the HRZ lumping and full mass matrix, the wave-front is slightly delayed 82 when compared with the analytical solution. 83 84 \begin{figure}[ht] 85 gross 3379 \centerline{\includegraphics[width=7cm]{lumping_valet_a_1}} 86 \caption{Amplitude at point $(\frac{1}{2},\frac{1}{2})$ using the accelaraton formulation~\ref{LUMPING WAVE VALET 2} of the 87 Velet scheme with order $1$ elements, element size $dx=0.01$, and $c=1$.} 88 \label{FIG LUMPING VALET A} 89 \end{figure} 90 91 ahallam 3381 \begin{figure}[ht] 92 gross 3379 \centerline{\includegraphics[width=7cm]{lumping_valet_a_2}} 93 \caption{Amplitude at point $(\frac{1}{2},\frac{1}{2})$ using the accelaraton formulation~\ref{LUMPING WAVE VALET 2} of the 94 Velet scheme with order $2$ elements, element size $0.01$, and $c=1$.} 95 \label{FIG LUMPING VALET B} 96 \end{figure} 97 98 ahallam 3381 \begin{figure}[ht] 99 gross 3379 \centerline{\includegraphics[width=7cm]{lumping_valet_u_1}} 100 \caption{Amplitude at point $(\frac{1}{2},\frac{1}{2})$ using the displacement formulation~\ref{LUMPING WAVE VALET 3} of the 101 Velet scheme with order $1$ elements, element size $0.01$ and $c=1$.} 102 \label{FIG LUMPING VALET C} 103 \end{figure} 104 105 ahallam 3381 Alternatively, one can directly solve for $u^{(n)}$ by inserting 106 equation~\ref{LUMPING WAVE VALET} into equation~\ref{LUMPING WAVE VALET 2}: 107 gross 3379 \begin{eqnarray} \label{LUMPING WAVE VALET 3} 108 u^{(n)}=2u^{(n-1)}-u^{(n-2)} + (dt\cdot c)^2 u^{(n-1)}_{,ii} \; . 109 \end{eqnarray} 110 ahallam 3381 This can also be interpreted as a PDE that must be solved at each time-step, but 111 for the unknown $u^{(n)}$. 112 As per equation~\ref{LINEARPDE.SINGLE.1} we set the general form coefficients to: 113 $D=1$; $Y=2u^{(n-1)}-u^{(n-2)}$; and $X=-(h\cdot c)^2 u^{(n-1)}_{,i}$. 114 For the full mass matrix, the acceleration ~\ref{LUMPING WAVE VALET 2} and displacement formulations ~\ref{LUMPING WAVE VALET 3} 115 are identical. 116 gross 3379 117 ahallam 3381 The displacement solution is depicted in Figure~\ref{FIG LUMPING VALET C}. The 118 domain utilised order $1$ elements (for order $2$ both 119 lumping methods are unstable). The solutions for the exact and the full mass 120 matrix approximation are almost identical while the lumping solutions, whilst 121 identical to each other, exhibit a considerably faster wave-front propagration 122 and a decaying amplitude. 123 124 \subsection{Advection equation} 125 Consider now, a second example that demonstrates the advection equation 126 gross 3379 \begin{eqnarray} \label{LUMPING ADVECTIVE} 127 u_{,t}=(v_i u)_i \; . 128 \end{eqnarray} 129 ahallam 3381 where $v_i$ is a given velocity field. To simplify this example, set $v_i=(1,0)$ and 130 gross 3379 \begin{equation} \label{LUMPING ADVECTIVE TEST} 131 u(x,t)= 132 \left\{ 133 \begin{array}{cl} 134 1 & x_0 < t \\ 135 0 & x_0 \ge t 136 \end{array} 137 ahallam 3381 \right}. 138 gross 3379 \end{equation} 139 ahallam 3381 The solution scheme implemented, is the two-step Taylor-Galerkin scheme 140 \index{Taylor-Galerkin scheme} 141 gross 3379 (which is in this case equivalent to SUPG\index{SUPG}): 142 ahallam 3381 the incremental formulation is given as 143 gross 3379 \begin{eqnarray} \label{LUMPING SUPG 1} 144 du^{(n-\frac{1}{2})} = \frac{dt}{2} (v_i u^{(n-1)})_i \\ 145 du^{(n)} = dt (v_i (u^{(n-1)}+du^{(n-\frac{1}{2})}) )_i \\ 146 u^{(n)} = u^{(n)} + du^{(n)} 147 \end{eqnarray} 148 ahallam 3381 This can be reformulated to calculate $u^{(n)}$ directly: 149 gross 3379 \begin{eqnarray} \label{LUMPING SUPG 2} 150 u^{(n-\frac{1}{2})} = u^{(n-1)} + \frac{dt}{2} (v_i u^{(n-1)})_i \\ 151 u^{(n)} = u^{(n-1)} + dt (v_i u^{(n-\frac{1}{2})} )_i 152 \end{eqnarray} 153 ahallam 3381 In some cases it may be possible to combine the two equations to calculate 154 $u^{(n)}$ without the intermediate step. This approach is not discussed, because 155 it is inflexible when a greater number of terms (e.g. a diffusion term) are 156 added to the right hand side. 157 158 The advection problem is thus similar to the wave propagation problem, because 159 the time step also needs to satisfy the CFL condition 160 \index{Courant condition}\index{explicit scheme!Courant condition}. For the 161 advection problem, this takes the form 162 gross 3379 \begin{eqnarray} \label{LUMPING ADVECTION CFL} 163 dt = f \cdot \frac{dx}{\|v\|} . 164 \end{eqnarray} 165 ahallam 3381 where $dx$ is the mesh size and $f$ is a safty factor. 166 For this example, we again use $f=\frac{1}{6}$. 167 gross 3379 168 ahallam 3381 Figures~\ref{FIG LUMPING SUPG INC A} and~\ref{FIG LUMPING SUPG INC B} illustrate 169 the four incremental formulation solutions: the true solution; the exact mass matrix; 170 the HRZ lumping; and the row sum lumping. Observe, that for the order $1$ elements 171 case, there is little deviation from the exact solution before the wave front, 172 whilst there is a significant degree of osciallation after the wave-front has 173 passed. For the order $2$ elements example, all of the numerical techniques fail. 174 175 \begin{figure}[ht] 176 gross 3379 \centerline{\includegraphics[width=7cm]{lumping_SUPG_du_1}} 177 \caption{Amplitude at point $(\frac{1}{2},\frac{1}{2})$ using the incremental formulation~\ref{LUMPING SUPG 1} of the 178 Taylor-Galerkin scheme with order $1$ elements, element size $dx=0.01$, $v=(1,0)$.} 179 \label{FIG LUMPING SUPG INC A} 180 \end{figure} 181 182 ahallam 3381 \begin{figure}[ht] 183 gross 3379 \centerline{\includegraphics[width=7cm]{lumping_SUPG_du_2}} 184 \caption{Amplitude at point $(\frac{1}{2},\frac{1}{2})$ using the incremental formulation~\ref{LUMPING SUPG 1} of the 185 Taylor-Galerkin scheme with order $2$ elements, element size $0.01$, $v=(1,0)$.} 186 \label{FIG LUMPING SUPG INC B} 187 \end{figure} 188 189 ahallam 3381 Figure~\ref{FIG LUMPING SUPG A} depicts the results from the direct formulation 190 of the advection problem for an order $1$ mesh. Generally, the results have 191 improved when compared with the incremental formulation. The full mass matrix 192 still introduces some osciallation both before and after the arrival of the 193 wave-front at the observation point. The two lumping solutions are identical, and 194 have introduced additional smoothing to the solution. There are no oscillatory 195 effects when using lumping for this example. In Figure~\ref{FIG LUMPING SUPG Ab} 196 the mesh or element size has been reduced from 0.01 to 0.002 units. As predicted 197 by the CFL condition, this significantly improves the results when lumping is 198 applied. However, when utilising the full mass matrix, a smaller mesh size will 199 result in post wave-front oscilations which are higher frequency and slower to 200 decay. 201 gross 3379 202 ahallam 3381 Figure~\ref{FIG LUMPING SUPG B} illustrates the results when utilising elements 203 of order $2$. The full mass matrix and HRZ lumping formulations are unable to 204 correctly model the exact solution. Only the row sum lumping was capable of 205 producing a smooth and sensical result. 206 207 \begin{figure}[ht] 208 gross 3379 \centerline{\includegraphics[width=7cm]{lumping_SUPG_u_1}} 209 \caption{Amplitude at point $(\frac{1}{2},\frac{1}{2})$ using the direct formulation~\ref{LUMPING SUPG 2} of the 210 Taylor-Galerkin scheme using order $1$ elements, element size $dx=0.01$, $v=(1,0)$.} 211 \label{FIG LUMPING SUPG A} 212 \end{figure} 213 ahallam 3381 214 \begin{figure}[ht] 215 gross 3379 \centerline{\includegraphics[width=7cm]{lumping_SUPG_u_1b}} 216 \caption{Amplitude at point $(\frac{1}{2},\frac{1}{2})$ using the direct formulation~\ref{LUMPING SUPG 2} of the 217 Taylor-Galerkin scheme using order $1$ elements, element size $dx=0.002$, $v=(1,0)$.} 218 \label{FIG LUMPING SUPG Ab} 219 \end{figure} 220 221 ahallam 3381 \begin{figure}[ht] 222 gross 3379 \centerline{\includegraphics[width=7cm]{lumping_SUPG_u_2}} 223 \caption{Amplitude at point $(\frac{1}{2},\frac{1}{2})$ using the direct formulation~\ref{LUMPING SUPG 2} of the 224 Taylor-Galerkin scheme using order $2$ elements, element size $0.01$, $v=(1,0)$.} 225 \label{FIG LUMPING SUPG B} 226 \end{figure} 227 228 ahallam 3381 \subsection{Sumamry} 229 The examples in this section have demonstrated the capabilities and limitations 230 of both HRZ and row sum lumping with comparisons to the exact and full mass 231 matrix solutions. Wave propagation type problems that utilise lumping, produce 232 results simular the full mass matrix at significantly 233 lower computation cost. An accelleration based formulation, with HRZ lumping 234 should be implemented for such problems, and can be appied to both order $1$ and 235 order $2$ elements. 236 gross 3379 237 ahallam 3381 In transport type problems, it is essential that row sum lumping is used to 238 achieve a smooth solution. Additionally, it is not recommended that second order 239 elements be used in advection type problems. 240 gross 3379 241 242