An Unconditionally Stable Parallel Difference Scheme for Telegraph Equation

<table>The computational molecule for node (<svg height="18.362499" id="M85" style="vertical-align:-5.08192pt" version="1.1" viewbox="0 0 37.362499 18.362499" width="37.362499" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.625,11.413)"><path d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37
q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" id="x1D465"></path></g>
<g transform="matrix(.012,-0,0,-.012,10.125,15.5)"><path d="M425 682q0 -40 -49 -59q-36 -13 -110 -13q-31 -18 -52.5 -49.5t-21.5 -67.5q0 -34 17.5 -57.5t45.5 -28.5q63 15 97 15q18 0 18 -16q0 -13 -22 -36t-34 -23q-50 0 -93 8q-47 -20 -80.5 -66.5t-33.5 -96.5q0 -76 84 -103q33 -11 75 -21q101 -26 101 -93q0 -51 -39 -95.5
t-76 -68.5l-22 20q60 69 60 111q0 21 -14 34t-55 22q-21 4 -63 16q-68 19 -101.5 52.5t-33.5 96.5q0 125 157 206q-27 10 -46 36t-19 58q0 84 103 152q-43 3 -71.5 21t-28.5 45q0 37 65 59l17 -6q-9 -14 -9 -33q0 -21 17.5 -36t51.5 -19q82 53 147 53q18 0 18 -17z" id="x1D709"></path></g>
<g transform="matrix(.017,-0,0,-.017,16.075,11.413)"><path d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" id="x2C"></path></g><g transform="matrix(.017,-0,0,-.017,22.773,11.413)"><path d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z" id="x1D461"></path></g><g transform="matrix(.017,-0,0,-.017,28.569,11.413)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg>) (a) and node (<svg height="18.362499" id="M86" style="vertical-align:-5.08192pt" version="1.1" viewbox="0 0 50.0625 18.362499" width="50.0625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.625,11.413)"><path d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37
q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" id="x1D465"></path></g>
<g transform="matrix(.012,-0,0,-.012,10.125,15.5)"><path d="M425 682q0 -40 -49 -59q-36 -13 -110 -13q-31 -18 -52.5 -49.5t-21.5 -67.5q0 -34 17.5 -57.5t45.5 -28.5q63 15 97 15q18 0 18 -16q0 -13 -22 -36t-34 -23q-50 0 -93 8q-47 -20 -80.5 -66.5t-33.5 -96.5q0 -76 84 -103q33 -11 75 -21q101 -26 101 -93q0 -51 -39 -95.5
t-76 -68.5l-22 20q60 69 60 111q0 21 -14 34t-55 22q-21 4 -63 16q-68 19 -101.5 52.5t-33.5 96.5q0 125 157 206q-27 10 -46 36t-19 58q0 84 103 152q-43 3 -71.5 21t-28.5 45q0 37 65 59l17 -6q-9 -14 -9 -33q0 -21 17.5 -36t51.5 -19q82 53 147 53q18 0 18 -17z" id="x1D709"></path></g><g transform="matrix(.012,-0,0,-.012,15.456,15.5)"><path d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" id="x2B"></path></g><g transform="matrix(.012,-0,0,-.012,22.44,15.5)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
<g transform="matrix(.017,-0,0,-.017,28.775,11.413)"><path d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" id="x2C"></path></g><g transform="matrix(.017,-0,0,-.017,35.473,11.413)"><path d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z" id="x1D461"></path></g><g transform="matrix(.017,-0,0,-.017,41.269,11.413)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg>) (b).</table>

Mathematical Problems in Engineering

fig2

Figure 2

Figure 2: An Unconditionally Stable Parallel Difference Scheme for Telegraph Equation 