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<p>Thank you Roland for your reply!</p>
<p>In the meanwhile I was able to solve the issue. I changed my
original evolution requation <br>
</p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mi>ζ</mi><mo>̈</mo></mover><mo>=</mo><mi>A</mi><msup><mi>ζ</mi><mo>′</mo></msup><mo>+</mo><mi>B</mi><msup><mi>ζ</mi><mo>″</mo></msup><mo>+</mo><mi>C</mi><mi>ζ</mi></mrow><annotation
encoding="TeX">\ddot{\zeta} = A \zeta' + B \zeta'' +C \zeta</annotation></semantics></math>
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics>
</semantics></math>into a system (1) of <br>
</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover><mi>ζ</mi><mo>˙</mo></mover><mo>=</mo><mi>Π</mi></mrow><annotation
encoding="TeX">\dot{\zeta} = \Pi</annotation></semantics></math><math
display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mi>Π</mi><mo>˙</mo></mover><mo>=</mo><mi>A</mi><msup><mi>ζ</mi><mo>′</mo></msup><mo>+</mo><mi>B</mi><msup><mi>ζ</mi><mo>″</mo></msup><mo>+</mo><mi>C</mi><mi>ζ</mi></mrow><annotation
encoding="TeX">\dot{\Pi}=A\zeta'+B\zeta''+C\zet</annotation></semantics></math>
I wanted to avoid the second-order numerical derivatives by using
a system (2)<br>
</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover><mi>ζ</mi><mo>˙</mo></mover><mo>=</mo><mi>Π</mi></mrow><annotation
encoding="TeX">\dot{\zeta}=\Pi</annotation></semantics></math><math
display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mi>Π</mi><mo>˙</mo></mover><mo>=</mo><mi>A</mi><mi>H</mi><mo>+</mo><mi>B</mi><msup><mi>H</mi><mo>′</mo></msup><mo>+</mo><mi>C</mi><mi>ζ</mi></mrow><annotation
encoding="TeX">\dot{\Pi}=AH+BH'+C\zeta</annotation></semantics></math><math
display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover><mi>H</mi><mo>˙</mo></mover><mo>=</mo><msup><mi>Π</mi><mo>′</mo></msup></mrow><annotation
encoding="TeX">\dot{H}=\Pi'</annotation></semantics></math>which
was source of the problem. This system is analytically equivalent
to the one above, but indeed not strongly hyperbolic. <br>
</p>
<p>I also had to change the boundaries near the r=0 region of the
grid, but the main issue was the aforementioned one. System (1)
works with the new boundary conditions.<br>
</p>
<div class="moz-cite-prefix">Thank you very much!</div>
<div class="moz-cite-prefix">Best regards,</div>
<div class="moz-cite-prefix">Severin<br>
</div>
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